We study the semi-classical tunneling of scalar and fermion fields from the horizon of a Constant Curvature Black Hole, which is locally AdS and whose five-dimensional analogue is dual to ${\cal N}=4$ Super Yang-Mills. In particular, we highlight the strong reliance of the tunneling method for Hawking radiation on near-horizon symmetries, a fact often hidden behind the algorithmic procedure with which the tunneling approach tends to be used. We ultimately calculate the emission rate of scalars and fermions, and hence the black hole's Hawking temperature.
In this paper, we investigate the tunneling process of charged massive bosons $W^{\pm }$ (spin-1 particles) from noncommutative charged black holes such as charged RN black holes and charged BTZ black holes. By applying the WKB approximation and by using the Hamilton-Jacobi equation we derive the tunneling rate and the corresponding Hawking temperature for those black holes configuration. Furthermore, we show the quantum gravity effects using the GUP on the Hawking temperature for the noncommutative RN black holes. The tunneling rate shows that the radiation deviates from pure thermality and i…
We present a derivation of Hawking radiation through tunneling mechanism for a general class of asymptotically flat, spherically symmetric spacetimes. The tunneling rate $Γ\sim \exp{(ΔS)}$ arises as a consequence of the first law of thermodynamics, TdS=dE + PdV. Therefore, this approach demonstrates how tunneling is intimately connected with the first law of thermodynamics through the principle of conservation of energy. The analysis is also generally applicable to any reasonable theory of gravity so long as the first law of thermodynamics for horizons holds in the form, TdS=dE + PdV.
We develop the argument that initial real tunneling in quantum gravity be contemplated as a thermodynamical analogous to a black hole condensate in equilibrium with Hawking's radiation in a box. The total entropy is always maximized in the Lorentzian sector of the manifold, and, in this sense, tunneling is predicted.
In this paper, a preliminary correspondence between the thermodynamic curvature and the isoperimetric theorem is established from a $4$-dimensional ultraspinning black hole. We find that the thermodynamic curvature of ultraspinning black hole is negative which means the ultraspinning black hole is likely to present an attractive between its molecules phenomenologically if we accept the analogical observation that the thermodynamic curvature reflects the interaction between molecules in a black hole system. Meanwhile we obtain a general conclusion that the thermodynamic curvature of the extreme…
The tunneling formalism in the Hamilton-Jacobi approach is adopted to study Hawking radiation of massless Dirac particles from spherically symmetric black hole spacetimes incorporating the effects of the generalized uncertainty principle. The Hawking temperature is found to contain corrections from the generalized uncertainty principle. Further, we show from this result that the ratio of the GUP corrected energy of the particle to the GUP corrected Hawking temperature is equal to the ratio of the corresponding uncorrected quantities. This result is then exploited to compute the Hawking tempera…
We investigate the thermodynamic curvature in the Hawking-Page phase transition and the second-order phase transition of the AdS black hole. It is shown that the thermodynamic curvature has the same behavior in these two different phase transitions. Specifically, the thermodynamic curvature in the Hawking-Page phase transition is the power function of the Hawking-Page phase transition temperature with exponent related to the spacetime dimension, or is invariably proportional to the reciprocal of the Hawking-Page phase transition entropy. For the thermodynamic curvature in the second-order phas…
In this paper, we visit the thermodynamic criticality and thermodynamic curvature of the charged AdS black hole in a new phase space. It is shown that when the square of the total charge of the charged black hole is considered as a thermodynamic quantity, the charged AdS black hole also admits an van der Waals-type critical behavior without the help of thermodynamic pressure and thermodynamic volume. Based on this, we study the fine phase structures of the charged AdS black hole with fixed AdS background in the new framework. On the one hand, we give the phase diagram structures of the charged…
Some thermodynamic properties of the Bañados-Teitelboim-Zanelli (BTZ) black hole are studied to get the effective dimension of its corresponding statistical model. For this purpose, we make use of the geometrical approach to the thermodynamics: Considering the black hole as a thermodynamic system with two thermodynamic variables (the mass $M$ and the angular momemtum $J$), we obtain two-dimensional Riemannian thermodynamic geometry described by positive definite Ruppeiner metric. From the thermodynamic curvature we find that the extremal limit is the critical point. The effective spatial dimen…
In modified theories of gravity, higher curvature terms may be added to the Einstein-Hilbert action. Conventionally, the effects of the higher curvature terms on the black hole thermodynamics are rather difficult to obtain. In this paper, we show that, at least at the first order level, the corrections to the thermodynamics of the Schwarzschild black hole can be easily obtained, without solving the modified black hole metric. We examine some specific examples, which produces the known results in the literature, as well as those previously unknown. Furthermore, we find some nonperturbative exac…
We compute the Euclidean action for constant curvature black holes (CCBHs), as an attempt to associate thermodynamic quantities to these solutions of Einstein anti-de Sitter (AdS) gravity. CCBHs are gravitational configurations obtained by identifications along isometries of a $D$-dimensional globally AdS space, such that the Riemann tensor remains constant. Here, these solutions are interpreted as extended objects, which contain a $(D-2)$-dimensional de-Sitter brane as a subspace. Nevertheless, the computation of the free energy for these solutions shows that they do not obey standard thermod…
We show that the recent tunneling formulas for black hole radiation in static, spherically symmetric spacetimes follow as a consequence of the first law of black hole thermodynamics and the area-entropy relation based on the radiation temperature. A tunneling formula results even if the radiation temperature is different from the one originally derived by Hawking and this is discussed in the context of the recent factor of 2 problem. In particular, it is shown that if the radiation temperature is higher than the Hawking temperature by a factor of two, thermodynamics then leads to a tunneling f…
We investigate the tunneling phenomenon of particles through the horizon of a magnetized Ernst-like black hole. We employ the modified Lagrangian equation with the extended uncertainty principle for this black hole. We determine a tunneling rate and the related Hawking temperature for this black hole by using the WKB approach in the field equation. In addition, we examine the graph behavior of the Hawking temperature in relation to the black hole event horizon. We explore the stability analysis of this black hole by taking into account the impact of quantum gravity on Hawking temperatures. The…
Due to the asymptotic structure of the black hole solution, there are two different thermodynamic schemes for the charged Banados-Teitelboim-Zanelli (BTZ) black hole. In one scheme, the charged BTZ black hole is super-entropic, while in the other, it is not (the reverse isoperimetric inequality is saturated). In this paper, we investigate the thermodynamic curvature of the charged BTZ black hole in different coordinate spaces. We find that in both schemes, the thermodynamic curvature is always positive, which may be related to the information of repulsive interaction between black hole molecul…
We give a new and conceptually simple approach to obtain the first law of black hole thermodynamics from a basic thermodynamical property that entropy (S) for any stationary black hole is a state function implying that dS must be an exact differential. Using this property we obtain some conditions which are analogous to Maxwell's relations in ordinary thermodynamics. From these conditions we are able to explicitly calculate the semiclassical Bekenstein-Hawking entropy, considering the most general metric represented by the Kerr-Newman spacetime. We extend our method to find the corrected entro…
The relationship between thermodynamics and the Lloyd bound on the holographic complexity for a black hole has been of interest. We consider $D$ dimensional anti-de Sitter black holes with hyperbolic geometry as well as black holes with momentum relaxation that have a minimum for temperature and mass. We show that the singular points of the thermodynamic curvature of the black holes, as thermodynamic systems, correspond to the zero points of the action and volume complexity at the Lloyd bound. For such black holes with a single horizon, the complexity of volume and the complexity of action at …
We investigate the intrinsic and extrinsic curvatures of certain hypersurfaces in the thermodynamic geometry of a physical system and show that they contain useful thermodynamic information. For an anti-Reissner-Nordström-(A)de Sitter black hole (Phantom), the extrinsic curvature of a constant $Q$ hypersurface has the same sign as the heat capacity around the phase transition points. For a Kerr-Newmann-AdS (KN-AdS) black hole, the extrinsic curvature of $Q \to 0$ hypersurface (Kerr black hole) or $J \to 0$ hypersurface (RN black black hole) has the same sign as the heat capacity around the pha…
We apply the tunneling process of charged massive particles through the quantum horizon of a Reissner-Nordstrom black hole in a new noncommutative gravity scenario. In this model, the tunneling amplitude on account of noncommutativity influences in the context of coordinate coherent states is modified. Our calculation points out that the emission rate satisfies the first law of black hole thermodynamics and is consistent with an underlying unitary theory.
We resolve the controversial issue of the geometric approach to the black hole thermodynamics. The geometric description of the equilibrium thermodynamics comes from Ruppeiner geometry based on a metric on the thermodynamic state space. For this purpose, we consider the Reissner-Nordström-AdS (RN-AdS) black hole which provides two different ensembles: canonical ensemble for fixed-charge case and grand canonical ensemble for fixed-potential case. Two cases are independent and cannot be mixed into each other. Hence, we calculate different Ruppeiner curvatures for two ensembles. However, we could…
We study the thermodynamics of some cosmological models based on modified gravity, a braneworld with induced gravity and curvature effect. Dark energy component seems necessary if the models are to approach thermal equilibrium in the long run.
We present a systematic and consistent construction of geometrothermodynamics by using Riemannian contact geometry for the phase manifold and harmonic maps for the equilibrium manifold. We present several metrics for the phase manifold that are invariant with respect to Legendre transformations and induce thermodynamic metrics on the equilibrium manifold. We review all the known examples in which the curvature of the thermodynamic metrics can be used as a measure of the thermodynamic interaction.
In this paper, we study the thermodynamic quantities of Friedmann-Robertson-Walker (FRW) universe by using the tunneling formalism beyond semiclassical approximation developed by \emph{Banerjee} and \emph{Majhi}\cite{beyond0}. For this we first calculate the corrected Hawking-like temperature on apparent horizon by considering both scalar particle and fermion tunneling. With this corrected Hawking-like temperature, the explicit expressions of the corrected entropy of apparent horizon for various gravity theories including Einstein gravity, Gauss-Bonnet gravity, Lovelock gravity, $f(R)$ gravity…
In the space of thermodynamic equilibrium states we introduce a Legendre invariant metric which contains all the information about the thermodynamics of black holes. The curvature of this thermodynamic metric becomes singular at those points where, according to the analysis of the heat capacities, phase transitions occur. This result is valid for the Kerr-Newman black hole and all its special cases and, therefore, provides a unified description of black hole phase transitions in terms of curvature singularities.
The $P$-$V$ phase transition and critical behavior in the extended phase space of asymptotic Anti-de Sitter (AdS) black holes have been widely investigated, in which four critical exponents around critical point are found to be consistent with values in the mean field theory. Recently, another critical exponent $ν$ related to divergent correlation length at critical point is proposed by using thermodynamic curvature scalar $R_N$ in the charged AdS black hole. In this paper, we develop a method to investigate the divergent behavior of $R_N$ at critical point, and find that the divergent behavio…
Massive charged and uncharged particles tunneling from commutative Reissner-Nordstrom black hole horizon has been studied with details in literature. Here, by adopting the coherent state picture of spacetime noncommutativity, we study tunneling of massive and charged particles from a noncommutative inspired Reissner-Nordstrom black hole horizon. We show that Hawking radiation in this case is not purely thermal and there are correlations between emitted modes. These correlations may provide a solution to the information loss problem. We also study thermodynamics of noncommutative horizon in thi…
We study the thermodynamical aspects of $f(R)$ gravity in the Jordan and the Einstein frame, and we investigate the corresponding equivalence of the thermodynamical quantities in the two frames. We examine static spherically symmetric black hole solutions with constant Ricci scalar curvature $R$, and as we demonstrate, the thermodynamical quantities in the two frames are equivalent. However, for the case of black holes with non-constant scalar curvature $R$, the thermodynamical equivalence of the two frames is no longer valid. In addition, we examine cosmological solutions with non-trivial cur…
We study the AdS rotating black hole solution for the Bergshoeff-Hohm-Townsend (BHT) massive gravity in three dimensions. The field equations of the asymptotically AdS black hole of the static metric can be expressed as the first law of thermodynamics, i.e. $dE=TdS-PdV$. The corrected Hawking-like temperature and entropy of asymptotically AdS rotating black hole are calculated using the Cardy formula and the tunneling method. Comparison of these methods will help identify the unknown leading correction parameter $β_1$ in the tunneling method.
We present exact analytical black hole solutions with conformal anomaly in AdS space and discuss the thermodynamical properties of these black hole solutions. These black holes can have a positive, zero and negative constant curvature horizon, respectively. For the black hole with a positive constant curvature horizon, there exists a minimal horizon determined by the coefficient of the trace anomaly, the black hole with a smaller horizon is thermodynamically unstable, while it is stable for the case with a larger horizon. The Hawking-Page transition happens in this case. For the black hole wit…
In this paper, we study the effect of dark energy on the extended thermodynamic structure and interacting microstructures of black holes in AdS, through an analysis of thermodynamic geometry. Considering various limiting cases of the novel equation of state obtained in charged rotating black holes with quintessence, and taking enthalpy $H$ as the key potential in the extended phase space, we scrutinize the behavior of the Ruppeiner curvature scalar $R$ in the entropy-pressure $(S,P)$-plane (or equivalently in the temperature-volume ($T,V$)-plane). Analysis of $R$ empirically reveals that dark …
We study the tunneling of massless scalars across black hole horizons in any number of spacetime dimensions greater than three. Our analysis finds that corrections due to backreaction and the inverse dimensional expansion are naturally concomitant, and furnishes a simple proof of the classic relation between entropy and area in all spacetime dimensions, finite or infinite. We conclude with a discussion of the limit in which the the number of spacetime dimensions is taken to infinity, where we find that thermodynamic quantities are related to the "thickness" of the membrane on which all the cur…
In the AdS/CFT correspondence, a dynamical cosmological constant $Λ$ in the bulk corresponds to varying the number of colors $N$ in the boundary gauge theory with a chemical potential $μ$ as its thermodynamic conjugate. In this work, within the context of Schwarzschild black holes in $AdS_5 \times S^5$ and its dual finite temperature $\mathcal{N}=4$ superconformal Yang-Mills theory at large $N$, we investigate thermodynamic geometry through the behavior of the Ruppeiner scalar $R$. The sign of $R$ is an empirical indicator of the nature of microscopic interactions and is found to be negative f…
Due to the non-independence of entropy and thermodynamic volume for spherically symmetric black holes in the AdS spacetime, when applying the Ruppeiner thermodynamic geometry theory to these black holes, we often encounter an unavoidable problem of the singularity about the line element of thermodynamic geometry. In this paper, we propose a basic and natural scheme for dealing with the thermodynamic geometry of spherically symmetric AdS black holes. We point out that enthalpy, not internal energy, is the fundamental thermodynamic characteristic function for the Ruppeiner thermodynamic geometry…
Some thermodynamic quantities of nonrelativistic ideal boson and fermion gases in the static Taub universe are derived to first order in a small anisotropy parameter d which measuring the deformation from the spherical Einstein universe. They are used to investigate the problem of how the curvature anisotropy affects the thermodynamic behaviors of an ideal gas. It is found that, when the universe is in the oblate configuration (i.e., d > 0), the effect of curvature anisotropy is to increase the number of the fraction in the Bose-Einstein condensation and to decrease the fermion distribution fu…
Employing the thermodynamic geometry approach, we explore phase transition of four dimensional spinning black holes in an anti-de Sitter (AdS) spaces and found the following novel results. (i) Contrary to the charged AdS black hole, thermodynamic curvature of the spinning AdS black hole diverges at the critical point, without needing normalization.(ii) There is a certain region with small entropy in the space of parameters for which the thermodynamic curvature is positive and the repulsive interaction dominates. Such behavior exists even when the pressure is extremely large. (iii) The dominant…
We study thermodynamics and thermodynamic geometry of a five-dimensional Schwarzschild AdS black hole in $AdS_5\times{S^5}$ spacetime by treating the cosmological constant as the number of colors in the boundary gauge theory and its conjugate quantity as the associated chemical potential. It is found that the chemical potential is always negative in the stable branch of black hole thermodynamics and it has a chance to be positive, but appears in the unstable branch. We calculate scalar curvatures of the thermodynamical Weinhold metric, Ruppeiner metric and Quevedo metric, respectively and we f…
There has been much work on explaining Hawking radiation as a quantum tunneling process through horizons. Basically, this intuitive picture requires the calculation of the imaginary part of the action for outgoing particle. And two ways are known for achieving this goal: the null-geodesic method and the Hamilton-Jacobi method. We apply these methods to the charged black holes in 2D dilaton gravity which is originated from the low energy effective theory of type 0A string theory. We derive the correct Hawking temperature of the black holes including the effect of the back reaction of the radiat…
Thermodynamic geometry is applied to the Born-Infeld-anti-de Sitter black hole (BIAdS) in the four dimensions, which is a nonlinear generalization of the Reissner-NorstrÄom-AdS black hole (RNAdS). We compute the Weinhold as well as the Ruppeiner scalar curvature and find that the singular points are not the same with the ones obtained using the heat capacity. Legendre-invariant metric proposed by Quevedo and the metric obtained by using the free energy as the thermodynamic potential are obtained and the corresponding scalar curvatures diverge at the Davies points.
The geometry of thermodynamic state space is studied for asymptotically anti-de Sitter black holes in D-dimensional space times. Convexity of thermodynamic potentials and the analytic structure of the response functions is analysed. The thermodynamic potentials can be used to define a metric on the space of thermodynamic variables and two commonly used such metrics are the Weinhold metric, derived from the internal energy, and the Ruppeiner metric, derived from the entropy. The intrinsic curvature of these metrics is calculated for charged and for rotating black holes and it is shown that the …
In this paper, we consider a static black hole in $f(R)$ gravity. We recapitulate the expression for corrected thermodynamic entropy of this black hole due to small fluctuations around equilibrium. Also, we study the geometrothermodynamics (GTD) of this black hole and investigate the adaptability of the curvature scalar of geothermodynamic methods with phase transition points of this black hole. Moreover, we study the effect of correction parameter on thermodynamic behaviour of this black hole. We observe that the singular point of the curvature scalar of Ruppeiner metric coincides completely …
I give a relatively broad survey of thermodynamic curvature $R$, one spanning results in fluids and solids, spin systems, and black hole thermodynamics. $R$ results from the thermodynamic information metric giving thermodynamic fluctuations. $R$ has a unique status in thermodynamics as being a geometric invariant, the same for any given thermodynamic state. In fluid and solid systems, the sign of $R$ indicates the character of microscopic interactions, repulsive or attractive. $|R|$ gives the average size of organized mesoscopic fluctuating structures. The broad generality of thermodynamic pri…
The recently proposed restricted phase space thermodynamics is shown to be applicable to a large class of higher dimensional higher curvature gravity models coupled to Maxwell field, which are known as black hole scan models and are labeled by the spacetime dimension $d$ and the highest order $k$ of the Lanczos-Lovelock densities appearing in the action. Three typical example cases with $(d,k)=(5,1), (5,2)$ and $(6,2)$ are chosen as example cases and studied in some detail. These cases are representatives of Einstein-Hilbert, Chern-Simons and Born-Infield like gravity models. Our study indicat…
Thermodynamics unavoidably contains fluctuation theory, expressible in terms of a unique thermodynamic information metric. This metric produces an invariant thermodynamic Riemannian curvature scalar $R$ which, in fluid and spin systems, measures interatomic interactions. Specifically, $|R|$ measures the size of organized fluctuating microscopic structures, and the sign of $R$ indicates whether the interactions are effectively attractive or repulsive. $R$ has also been calculated for black hole thermodynamics for which there is no consensus about any underlying microscopic structures. It is hop…
From the point of view of tunneling, the physical meaning of the Davies critical point of a second order phase transition in the black hole thermodynamics is clarified. At the critical point, the nonthermal contribution vanishes so that the black hole radiation is entirely thermal. It separates two phases: one with radiation enhanced by the nonthermal contribution, the other suppressed by the nonthermal contribution. We show this in both charged and rotating black holes. The phase transition is also analyzed in the cases in which emissions of charges and angular momenta are incorporated.
We examine the thermodynamics of a new class of asymptotically AdS black holes with non-constant curvature event horizons in Gauss-Bonnet Lovelock gravity, with the cosmological constant acting as thermodynamic pressure. We find that non-trivial curvature on the horizon can significantly affect their thermodynamic behaviour. We observe novel triple points in 6 dimensions between large and small uncharged black holes and thermal AdS. For charged black holes we find a continuous set of triple points whose range depends on the parameters in the horizon geometry. We also find new generalizations o…
We study the thermodynamics and thermodynamic geometry of a five-dimensional Reissner-Nordström-AdS black hole in the extended phase space by treating the cosmological constant as being related to the number of colors in the boundary gauge theory and its conjugate quantity as the associated chemical potential. It is found that the contribution of the charge of the black hole to the chemical potential is always positive and the existence of charge make the chemical potential become positive more easily. We calculate the scalar curvatures of the thermodynamical Weinhold metric, Ruppeiner metric …
We apply the formalism of geometrothermodynamics to the case of black holes with cosmological constant in four and higher dimensions. We use a thermodynamic metric which is invariant with respect to Legendre transformations and determines the geometry of the space of equilibrium states. For all known black holes in higher dimensions, we show that the curvature scalar of the thermodynamic metric in all the cases is proportional to the heat capacity. As a consequence, phase transitions, which correspond to divergencies of the heat capacity, are represented geometrically as true curvature singula…
A real tunneling solution is an instanton for the Hartle-Hawking path integral with vanishing extrinsic curvature (vanishing ``momentum'') at the boundary. Since the final momentum is fixed, its conjugate cannot be specified freely; consequently, such an instanton will contribute to the wave function at only one or a few isolated spatial geometries. I show that these geometries are the extrema of the Hartle-Hawking wave function in the semiclassical approximation, and provide some evidence that with a suitable choice of time parameter, these extrema are the maxima of the wave function at a fix…
We study the Hawking radiation as charged particles' tunneling across the horizons of the Hot-NUT-Kerr-Newman-Kasuya spacetime by considering the spacetime background as dynamical and incorporating the self-gravitation effect of the emitted particles when the energy conservation, the angular momentum conservation, and the electric charge conservation are taken into account. Our result shows that the tunneling rate is related to the change of Bekenstein-Hawking entropy and the radiant spectrum is not pure thermal, but is consistent with an underlying unitary theory. The emission process is a re…
We disclose the deep connection between the thermodynamics and gravity in a general braneworld model with curvature correction terms on the brane and in the bulk, respectively. We show that the Friedmann equations on the 3-brane embedded in the 5D spacetime with curvature correction terms can be written directly in the form of the first law of thermodynamics on the apparent horizon. Using the first law, we extract the entropy expression of the apparent horizon on the brane, which is useful in studying the thermodynamical properties of the black hole horizon on the brane in Gauss-Bonnet gravity…
In this paper, we compute the Hawking temperature by applying quantum tunneling approach for the Horndeski like black holes. We utilize the semi-classical phenomenon and WKB approximation to the Lagrangian field equation involving generalized uncertainty principle (GUP) and compute the tunneling rate as well as Hawking temperature. For the zero gravity parameter, we obtain results consistent without correction parameter or original tunneling. Moreover, we study the thermal fluctuations of the considered geometry and examine the stable state of the system by heat capacity technique. We also inv…
Considering de Rham-Gabadadze-Tolley theory of massive gravity coupled with (ghost free) higher curvature terms arisen from the Lovelock Lagrangian, we obtain charged AdS black hole solutions in diverse dimensions. We compute thermodynamic quantities in the extended phase space by considering the variations of the negative cosmological constant, Lovelock coefficients ($α_{i}$) and massive couplings ($c_{i}$), and prove that such variations is necessary for satisfying the extended first law of thermodynamics as well as associated Smarr formula. In addition, by performing a comprehensive thermal…
I consider the problem of reconciling "tunneling" approaches to black-hole radiation with the treatment by quantum field theory in curved space-time. It is not possible to do this completely, but using what appears to be the most direct and natural correspondence, the simplest case of such an approach does not describe a tunneling process but rather a WKB approximation to a two-point function. What it computes is a sort of rescaled Unruh temperature associated with world-lines tracing Killing trajectories near the horizon. This temperature is numerically equal to the Hawking temperature, but a…
The work provides a possible explanation of a well motivated question - why the present universe is practically free from any noticeable footmarks of higher rank antisymmetric tensor fields, despite having the signatures of scalar, vector, fermion as well as symmetric rank 2 tensor field in the form of gravity ? The explanation proposed here originates from the higher curvature degrees of freedom present in a $F(R)$ gravity model. In such a model, we show that the scalar degree of freedom (also known as scalaron) associated with the higher curvature term may undergo a quantum tunneling which i…
In this paper we show that the vanishing of the scalar curvature of Ruppeiner-like metrics does not characterize the ideal gas. Furthermore, we claim through an example that flatness is not a sufficient condition to establish the absence of interactions in the underlying microscopic model of a thermodynamic system, which poses a limitation on the usefulness of Ruppeiner's metric and conjecture. Finally, we address the problem of the choice of coordinates in black hole thermodynamics. We propose an alternative energy representation for Kerr-Newman black holes that mimics fully Weinhold's approa…
