First Law Of Black Hole Thermodynamics Research Articles

Thermodynamics of charged three-dimensional scalar-tensor black holes: the problem of mathematical inconsistency and its solution

By applying conformal transformations on the action of scalar–tensor-Euler–Heisenberg theory, we obtain the exact black hole (BH) solutions in its conformal related, the well-known Einstein frame. Through imposing the conditions of (a) vanishing the electric potential at large distance from the source and (b) validity of the first law of BH thermodynamics, we obtain a set of three requirements which are not consistent, mathematically. For solving this problem we assume that under conformal transformations the nonlinearity parameter of Euler–Heisenberg (EH) electrodynamics must transform as a→ae4αϕ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$a\\rightarrow ae^{4\\alpha \\phi }$$\\end{document}. Then, we obtain the exact solutions of this theory, in both of Einstein and Jordan frames, without any mathematical problems. After calculating thermodynamic quantities, we investigate validity of the thermodynamical first law (TFL) and thermal stability of the EH-BTZ BHs in both of Jordan and Einstein frames, separately.

Thermodynamics of novel scalar–tensor-Born–Infeld black holes

Two novel classes of four-dimensional exact black hole (BH) solutions have been obtained in the scalar–tensor (ST) theory which are coupled to Born–Infeld (BI) electrodynamics. To this end, a conformal transformation (CT) has been applied which transforms the action of ST–BI gravity to that of Einstein–dilaton–BI theory. The scalar-coupled BI theory, which has been introduced here, slightly differs from those have been used, previously. The analytical solutions have been obtained in the Einstein frame (EF) and two classes of charged dilatonic BHs, with unusual asymptotic behaviors, have been presented. All the solutions coincide with the corresponding values of Einstein–dilaton–Maxwell theory, in the limit of large BI parameter. By calculating thermodynamic parameters and, noting the Smarr mass relation, we showed that the first law of BH thermodynamics (FLT) is valid for the novel dilatonic BHs. Stability of the BHs has been investigated in EF, making use of the canonical ensemble method and noting the signature of the BH heat capacity (HC). Next, by use of the inverse CT, the solutions of ST theory have been obtained from their EF counterparts. Although, the entropy of ST BHs violates entropy-area law, the thermodynamic and conserved quantities have been obtained noting their conformal invariance property. It has been found that the ST BHs have the same thermodynamic and stability properties as the Einstein–dilaton ones.

Thermodynamics of higher-dimensional Brans-Dicke black holes in the presence of a conformal-invariant field inspired by power-Maxwell electrodynamics

Abstract By use of the conformal transformations, in addition to translating the Brans-Dicke (BD) action to the Einstein frame (EF), we introduce an electromagnetic Lagrangian which preserves conformal invariance. We solve the EF field equations, which mathematically are confronted with the problem of indeterminacy, by use of an exponential ansatz function. When the self-interacting potential is absent or is taken constant in the BD action, the exact solution of this theory is just that of Einstein-conformal-invariant theory with a trivial scalar field. This is higher-dimensional (HD) analogues of the same considered in [1]. The EF general solution admits two classes of black holes (BHs) with non-flat and non-AdS asymptotic behavior which can produce extreme and multi-horizon ones. We obtain the exact solutions of BD-conformal-invariant theory, by applying inverse conformal transformations, which show two classes of extreme and multi-horizon BHs too. Based on the fact that thermodynamic quantities remain unchanged under conformal transformations, we show that the first law of BH thermodynamics is valid in the Jordan frame (JF). We analyze thermal stability of the HD BD-conformal-invariant BHs by use of the canonical ensemble method.

Black hole thermodynamics in the Brans–Dicke–Maxwell theory

It has been shown that the exact solutions of four-dimensional (4D) Brans–Dicke–Maxwell (BDM) theory is nothing other than Reissner–Nordström (RN) black hole (BH)s coupled to a trivial constant scalar field (Cai and Myung in Phys Rev D 56:3466, 1997). Here, we show that it is the case only when the scalar potential is taken constant or equal to zero. Then, through obtaining the exact solutions, in the presence of a scalar potential, we show that this theory admits two classes of novel BH solutions which have been affected by a nontrivial scalar hair. Due to conformal invariance of Maxwell’s electrodynamics, multi-horizon BHs can occur which implies the anti-evaporation quantum effect. Inclusion of the scalar hair makes the asymptotic behavior of the solutions non-flat and non-AdS. Our novel solutions recover the RN-AdS BHs when the scalar field is turned off. Thermodynamic quantities of the 4D BDM BHs have been calculated by use of the appropriate methods and under the influence of scalar field. Then, by use of a Smarr-type mass formula, it has been found that the first law of BH thermodynamics is valid for our novel BHs. Thermal stability of the BDM BHs has been analyzed by use of the canonical ensemble and geometrical methods, comparatively.

Wald entropy in Kaluza-Klein black holes

We study the thermodynamics of the 4-dimensional electrically charged black-hole solutions of the simplest 5-dimensional Kaluza-Klein theory using Wald’s formalism. We show how the electric work term present in the 4-dimensional first law of black-hole thermodynamics arises in the purely gravitational 5-dimensional framework. In particular, we find an interesting geometric interpretation of the 4-dimensional electrostatic potential similar to the angular velocity in rotating black holes. Furthermore, we show how the momentum map equation arises from demanding compatibility between the timelike Killing vector of the black-hole solution and the spatial Killing vector of the 5-dimensional background.

Three-dimensional black holes with scalar hair coupled to a Maxwell-like electrodynamics

By consideration of an Einstein-dilaton nonlinear charged gravitating system, it has been shown that this theory is confronted with the problem of indeterminacy. It means that the number of independent differential equations is one less than the number of unknowns. To overcome this problem, the power-law and exponential ansatz functions have been used, separately. Through solving the field equations, in the presence of a Coulomb-like electric field, it has been found that this theory includes two novel classes of charged black holes (BHs) with unusual asymptotic behavior, for each ansatz. It has been found that, under some circumstances, both of the ansatz functions lead to the same results. The novel exact solutions show BHs with one horizon, two horizons and without horizon. Using a Smarr-type mass formula, the validity of the first law of BH thermodynamics (FLT) has been proved, after calculating the thermodynamic and conserved quantities. Making use of thermodynamical and geometrical approaches, thermal stability of the BHs has been analyzed. Results of the aforementioned methods have been compared by use of the plots.

Thermodynamics of the Euler-Heisenberg-AdS black hole

We generalize the solution to the Euler-Heisenberg (EH) theory coupled to gravity that represents a nonlinearly charged static black hole (BH) introducing the cosmological constant; the obtained solution is characterized by four parameters: mass $M$, electric charge $Q$, cosmological constant $\Lambda$ (positive or negative) and the Euler-Heisenberg theory parameter $a$. Then we briefly analyze some BH features like horizons, electromagnetic field and geodesics. We mainly focus on its thermodynamic properties in the extended space where the anti-de Sitter parameter is interpreted as the pressure; we show the consistency between the Smarr formula and the first law of BH thermodynamics, interpreting the parameter of the EH theory as the vacuum polarization. We determine the equation of state and the critical points; the critical volume determines two branches of BHs, one near Maxwell behavior and a second one manifestly nonlinear electromagnetic. Moreover, the analysis of the Gibbs free energy indicates that two phase transitions can occur; we also construct the coexistence curve $P$-$T$ where the different phases of the BH can be observed; the critical point is characterized by the standard mean field theory exponents, and the critical variables satisfy $P_{\rm crit} v_{\rm crit}/ T_{\rm crit} = 3/8 $ plus terms of the ten thousandths order in the Euler-Heisenberg parameter $a$

\u03b1\u2032 corrections of Reissner-Nordstr\xf6m black holes

We study the first-order in α′ corrections to non-extremal 4-dimensional dyonic Reissner-Nordström (RN) black holes with equal electric and magnetic charges in the context of Heterotic Superstring effective field theory (HST) compactified on a T6. The particular embedding of the dyonic RN black hole in HST considered here is not supersymmetric in the extremal limit. We show that, at first order in α′, consistency with the equations of motion of the HST demands additional scalar and vector fields become active, and we provide explicit expressions for all of them. We determine analytically the position of the event horizon of the black hole, as well as the corrections to the extremality bound, to the temperature and to the entropy, checking that they are related by the first law of black-hole thermodynamics, so that ∂S/∂M = 1/T. We discuss the implications of our results in the context of the Weak Gravity Conjecture, clarifying that entropy corrections for fixed mass and charge at extremality do not necessarily imply corrections to the extremal charge-to-mass ratio.

Charged black holes in higher-dimensional Eddington-inspired Born-Infeld gravity

We study static (electrically)-charged solutions of Eddington-inspired Born-Infeld (EiBI) theory of gravity in general D-dimensional spacetime. We consider both linear (Maxwell) as well as nonlinear electrodynamics for the matter fields. In this particular work, the nonlinear theory we specifically consider is the Born-Infeld (BI) electrodynamics. The solutions describe higher-dimensional black holes in EiBI gravity. For the linear Maxwell field, we show that the electric field is still singular for D>4. This singularity is cured when EiBI is coupled to the BI electrodynamics. We obtain EiBI-BI black hole solutions in the limit of α˜≡4κb2/λ=1 and 2. We also investigate their thermodynamical property. We show that all solutions satisfy the first-law of black hole thermodynamics, from which their corresponding ADM mass can be extracted. It is found that κ imposes a charge screening that makes the corresponding Hawking temperature experiences some sudden jump from charged-type to the Schwarzschild-type at some critical value of κ. Thermodynamical stability reveals that the EiBI-BI black holes can exist with smaller horizon than their Reissner-Nordstrom (RN) counterparts.

Thermodynamic properties of Kehagias-Sfetsos black hole and KS/CFT correspondence

We speculate on various thermodynamic features of the inner horizon and outer horizons of Kehagias-Sfetsos (KS) black hole (BH) in the background of the Hořava-Lifshitz gravity. We compute particularly the area product, area sum, area minus and area division of the BH horizons. We find that they all are not showing universal behavior whereas the product is a universal quantity (Pradhan P., Phys. Lett. B, 747 (2015) 64). Based on these relations, we derive the area bound of all horizons. From the area bound we derive the entropy bound and irreducible mass bound for all the horizons . We also observe that the first law of BH thermodynamics and Smarr-Gibbs-Duhem relations do not hold for this BH. The underlying reason behind this failure is due to the scale invariance of the coupling constant. Moreover, we compute the Cosmic-Censorship-Inequality for this BH which gives the lower bound for the total mass of the spacetime and it is supported by the cosmic cencorship conjecture. Finally, we discuss the KS/CFT correspondence via a thermodynamic procedure.

Surface area products for Kerr-Taub-NUT space-time

We examine properties of the inner and outer horizon thermodynamics of Taub-NUT (Newman-Unti-Tamburino) and Kerr-Taub-NUT (KTN) black hole (BH) in four-dimensional Lorentzian geometry. We compare and contrasted these properties with the properties of Reissner Nordstrøm (RN) BH and Kerr BH. We focus on “area product”, “entropy product”, “irreducible mass product” of the event horizon and Cauchy horizons. Due to mass dependence, we speculate that these products have no nice quantization feature. Nor do they have any universal property. We further observe that the first law of BH thermodynamics and Smarr-Gibbs-Duhem relations do not hold for Taub-NUT (TN) and KTN BH in Lorentzian regime. The failure of these aforementioned features are due to the presence of the non-trivial NUT charge which makes the space-time be asymptotically non-flat, in contrast with RN BH and Kerr BH. Another reason for the failure is that Lorentzian TN and Lorentzian KTN geometries contain Dirac-Misner–type singularity, which is a manifestation of a non-trivial topological twist of the manifold. The black-hole mass formula and Christodoulou-Ruffini mass formula for TN and KTN BHs are also computed. These thermodynamic product formulae give us further understanding of the nature of inner as well as outer BH entropy at the microscopic level.

Area product and mass formula for Kerr–Newman–Taub–NUT spacetime

We derive area product, entropy product, area sum and entropy sum of the event horizon and Cauchy horizons for Kerr–Newman–Taub–NUT (Newman–Unti–Tamburino) black hole in four-dimensional Lorentzian geometry. We observe that these thermodynamic products are not universal (mass-independence) for this black hole (BH), whereas for Kerr–Newman (KN) BH such products are universal (mass-independence). We also examine the entropy sum and area sum. It is shown that they all depend on mass, charge and NUT parameter of the background spacetime. Thus, we can conclude that the universal (mass-independence) behavior of area product and entropy product, area sum and entropy sum for Kerr–Newman–Taub–NUT (KNTN) BH fails and which is also quite different from KN BH. We further show that the KNTN BH do not possess first law of BH thermodynamics and Smarr–Gibbs–Duhem relations, and that such relations are unlikely in the KN case. The failure of these aforementioned features are due to presence of the nontrivial NUT charge which makes the spacetime to be asymptotically non-flat, in contrast with KN BH. The other reason of the failure is that Lorentzian KNTN geometry contains Dirac–Misner type singularity, which is a manifestation of a nontrivial topological twist of the manifold. The BH mass formula and Christodoulou–Ruffini mass formula for KNTN BHs are also derived. Finally, we compute the area bound which is just Penrose like inequality for event horizon. From area bound we derive entropy bound. These thermodynamic products on the multi-horizon play a crucial role in BH thermodynamics to understand the microscopic nature of BH entropy.

Violation of the first law of black hole thermodynamics in f(T) gravity

We prove that, in general, the first law of black holethermodynamics, δQ = TδS, is violated in f(T) gravity.As a result, it is possible that there exists entropy production,which implies that the black hole thermodynamics can be innon-equilibrium even in the static spacetime. This feature is verydifferent from that of f(R) or that of other higher derivativegravity theories. We find that the violation of first law resultsfrom the lack of local Lorentz invariance in f(T) gravity. Byinvestigating two examples, we note that f′′(0) should be negativein order to avoid the naked singularities and superluminal motion oflight. When f′′(T) is small, the entropy of black holes in f(T)gravity is approximatively equal to f′(T)/4 A.

Lovelock black holes with maximally symmetric horizons

We investigate some properties of n( ⩾ 4)-dimensional spacetimes having symmetries corresponding to the isometries of an (n − 2)-dimensional maximally symmetric space in Lovelock gravity under the null or dominant energy condition. The well-posedness of the generalized Misner–Sharp quasi-local mass proposed in the past study is shown. Using this quasi-local mass, we clarify the basic properties of the dynamical black holes defined by a future outer trapping horizon under certain assumptions on the Lovelock coupling constants. The C2 vacuum solutions are classified into four types: (i) Schwarzschild–Tangherlini-type solution; (ii) Nariai-type solution; (iii) special degenerate vacuum solution; and (iv) exceptional vacuum solution. The conditions for the realization of the last two solutions are clarified. The Schwarzschild–Tangherlini-type solution is studied in detail. We prove the first law of black-hole thermodynamics and present the expressions for the heat capacity and the free energy.

Solitons and black holes in a generalized Skyrme model with dilaton-quarkonium field

Skyrme theory is among the viable effective theories which emerge from low-energy limit of quantum chromodynamics. Many of its generalizations include also a dilaton. Here we find new self-gravitating solutions, both solitons and black holes, in a Generalized Skyrme Model (GSM) in which a dilaton is present. The investigation of the properties of the solutions is done numerically. We find that the introduction of the dilaton in the theory does not change the picture qualitatively, only quantitatively. The model considered here has one free parameter more than the Einstein-Skyrme model which comes from the potential of the dilaton. We have applied also the turning point method to establish that one of the black-hole branches of solutions is unstable. The turning point method here is based on the first law of black-hole thermodynamics a detailed derivation of which is given in the Appendix of the paper.

Black-hole entropy and thermodynamics from symmetries

Given a boundary of spacetime preserved by a Diff(S1) sub-algebra, we propose a systematic method to compute the zero mode and the central extension of the associated Virasoro algebra of charges. Using these values in the Cardy formula, we may derive an associated statistical entropy to be compared with the Bekenstein–Hawking result. To illustrate our method, we study in detail the BTZ and the rotating Kerr–adS4 black holes (at spatial infinity and on the horizon). In both cases, we are able to reproduce the area law with the correct factor of 1/4 for the entropy. We also recover within our framework the first law of black-hole thermodynamics. We compare our results with the analogous derivations proposed by Carlip and others. Although similar, our method differs in the computation of the zero mode. In particular, the normalization of the ground state is automatically fixed by our construction.

The first law of black-hole thermodynamics for black holes in string theory

We investigate thermodynamical properties of four- and five-dimensional black-hole solutions of toroidally compactified string theory. We find an explicit expression for the first law of black-hole thermodynamics. We calculate the temperature T, angular velocity and the electromagnetic potentials on the horizon using two different methods.