A static spherically symmetric solution is obtained and examined in the framework of Einstein-power-Yang-Mills-dilaton theory. To derive this exact solution a dilaton potential $V(Φ)$ is taken into account. Thermodynamics of the black hole is also studied, namely we have obtained and investigated the temperature and heat capacity of the black hole which has allowed us characterize the stability of the black hole. Using the framework of the extended thermodynamic phase space we have derived the equation of state for the black hole. We have studied the Gibbs free energy of the black hole, which has the behaviour similar for other black holes with dilaton fields. To characterize the thermal behaviour of the black hole near the critical point we have written the Ehrenfest's equations and calculated Prigogine-Defay ratio.
We derive the global properties of static spherically symmetric solutions to the Einstein-Maxwell-dilaton system in the presence of an arbitrary exponential dilaton potential. We show that -- with the exception of a pure cosmological constant `potential' -- no asymptotically flat, asymptotically de Sitter or asymptotically anti-de Sitter solutions exist in these models.
A static black hole with spherical symmetry is obtained and examined in the framework of Einstein-Yang-Mills-dilaton theory. The obtained black hole solution allowed us to derive and investigate entropy, temperature and heat capacity. To better examine the thermodynamics of the black hole extended phase space is also used. On this ground the equation of state is obtained and studied. We have also investigated the Gibbs free energy and it is shown that below the critical temperature the system demonstrates phase transitions of the first as well as of the zeroth order which is notable feature fo…
In this work a static spherically symmetric black hole's solution within the Einstein-Maxwell-Yang-Mills-dilaton theory is derived. The obtained solution is examined, in particular, we have calculated the Kretschmann scalar which allowed us to characterize singularity points. Using the obtained black hole's solution we have studied its thermodynamics. Namely, the temperature was calculated, the first law was written, and the heat capacity was studied. The extended thermodynamics approach is utilized to obtain the equation of state. Within the extended thermodynamics the Gibbs free energy is de…
The energy associated with a static and spherically symmetric charged dilaton black hole is obtained for arbitrary value of the coupling parameter (which regulates the strength of the coupling of the dilaton to the Maxwell field) $β$. The energy distribution depends on $β$, whereas the total energy is independent of this and is given by the mass parameter of the black hole.
We calculate energy and momentum of a spherically symmetric dilaton frame using the gravitational energy-momentum 3-form within the tetrad formulation of general relativity (GR). The frame we use is characterized by an arbitrary function $Υ$ with the help of which all the previously found solutions can be reproduced. We show how the effect of inertia {\it (which is mainly reproduced from $Υ$)} makes the total energy and momentum always different from the well known result when we use the Riemannian connection ${{\widetilde Γ}_α}^β$. On the other hand, when use is made of the covariant formulat…
We investigate the static, spherically symmetric black hole solutions in the quasi-dilaton model and its generalizations, which are scalar extended dRGT massive gravity with a shift symmetry. We show that, unlike generic scalar extended massive gravity models, these theories do not admit static, spherically symmetric black hole solutions until the theory parameters in the dRGT potential is fine-tuned. When fine-tuned, the geometry of the static, spherically symmetric black hole is necessarily that of general relativity and the quasi-dilaton field is constant across the spacetime. The fine-tuni…
We classify the spherically symmetric solutions of the Einstein-Maxwell Dilaton field equations in D dimensions and find some exact solutions of the string theory at all orders of the string tension parameter. We also show the uniqueness of the black hole solutions of this theory in static axially symmetric spacetimes.
We consider an Einstein-Yang-Mills Lagrangian in a five dimensional space-time including a cosmological constant. Assuming all fields to be independent of the extra coordinate, a dimensional reduction leads to an effective (3+1)-dimensional Einstein-Yang-Mills-Higgs-dilaton model where the cosmological constant induces a Liouville potential in the dilaton field. We construct spherically symmetric solutions analytically in specific limits and study the generic solutions for vanishing dilaton coupling numerically. We find that in this latter case the solutions bifurcate with the branch of (Anti-…
A numerical study of static spherically symmetric monople solutions of a spontaneously broken SU(2) gauge theory coupled to a dilaton field is presented. Regular solutions seem to exist only up to a maximal value of the dilaton coupling. In addition to the generalization of the 't Hooft-Polyakov monopole a discrete family of regular solutions is found, corresponding to radial excitations, absent in the theory without dilaton.
We introduce a new attractor mechanism to find the entropy for spherically symmetric extremal black holes. The key ingredient is to find a two-dimensional (2D) dilaton gravity with the dilaton potential $V(ϕ)$. The condition of an attractor is given by $\nabla^2ϕ=V(ϕ_0)$ and $\bar{R}_2=-V^{\prime}(ϕ_0)$ and for a constant dilaton $ ϕ=ϕ_0$, these are also used to find the location of the degenerate horizon $r=r_{e}$ of an extremal black hole. As a nontrivial example, we consider an extremal regular black hole obtained from the coupled system of Einstein gravity and nonlinear electrodynamics. Th…
We discuss the static axially symmetric regular solutions, obtained recently in Einstein-Yang-Mills and Einstein-Yang-Mills-dilaton theory [1]. These asymptotically flat solutions are characterized by the winding number $n>1$ and the node number $k$ of the purely magnetic gauge field. The well-known spherically symmetric solutions have winding number $n=1$. The axially symmetric solutions satisfy the same relations between the metric and the dilaton field as their spherically symmetric counterparts. Exhibiting a strong peak along the $ρ$-axis, the energy density of the matter fields of the axi…
Analysing the static, spherically symmetric graviton-dilaton solutions in low energy string and Brans-Dicke theory, we find the following. For a charge neutral point star, these theories cannot predict non trivial PPN parameters, $β$ and $γ$, without introducing naked singularities. We then couple a cosmological constant $Λ$ as in low energy string theory. We find that only in low energy string theory, a non zero $Λ$ leads to a curvature singularity, which is much worse than a naked singularity. Requiring its absence upto a distance $r_*$ implies a bound $| Λ| < 10^{- 102} (\frac{r_*}{\rm pc})…
We consider the Einstein-Yang-Mills Lagrangian in a (4+n)-dimensional space-time. Assuming the matter and metric fields to be independent of the n extra coordinates, a spherical symmetric Ansatz for the fields leads to a set of coupled ordinary differential equations. We find that for n > 1 only solutions with either one non-zero Higgs field or with all Higgs fields constant exist. We construct the analytic solutions which fulfill this conditions for arbitrary n, namely the Einstein-Maxwell-dilaton solutions. We also present generic solutions of the effective 4-dimensional Einstein-Yang-Mills-…
We discuss the asymptotic form of the static axially symmetric, globally regular and black hole solutions, obtained recently in Einstein-Yang-Mills and Einstein-Yang-Mills-dilaton theory.
SU(3) Einstein-Yang-Mills-dilaton theory possesses sequences of static spherically symmetric sphaleron and black hole solutions for the SU(2) and the SO(3) embedding. The solutions depend on the dilaton coupling constant $γ$, approaching the corresponding Einstein-Yang-Mills solutions for $γ\rightarrow 0$, and Yang-Mills-dilaton solutions in flat space for $γ\rightarrow \infty$. The sequences of solutions tend to Einstein-Maxwell-dilaton solutions with different magnetic charges. The solutions satisfy analogous relations between the dilaton field and the metric for general $γ$. Thermodynamic p…
We study spherically and axially symmetric monopoles of the SU(2) Einstein-Yang-Mills-Higgs-dilaton (EYMHD) system with a new coupling between the dilaton field and the covariant derivative of the Higgs field. This coupling arises in the study of (4+1)-dimensional vortices in the Einstein-Yang-Mills (EYM) system.
The uniqueness of static spherically symmetric traversable wormholes with two asymptotically flat ends, subject to the higher-dimensional solutions of Einstein-Maxwell-phantom dilaton field equations was proved. We considered the case of an arbitrary dilaton coupling constant. Conformal positive energy theorem plays the key role in the considerations.
The trace anomaly for nonminimally coupled scalars in spherically reduced gravity obtained by Bousso and Hawking (hep-th/9705236) is incorrect. We explain the reasons for the deviations from our correct (published) result which is supported by several other recent papers.
We present arguments for the existence of a new type of solutions of the Euclidean Einstein-Yang-Mills-dilaton theory in $d=4$ dimensions. Possesing nonvanishing nonabelian charges, these nonselfdual configurations have no counterparts on the Lorentzian section. They provide, however, new saddle points in the Euclidean path integral.
We construct static and spherically symmetric particle-like and black hole solutions with magnetic or electric charge in the Einstein-Born-Infeld-dilaton system, which is a generalization of the Einstein-Maxwell-dilaton (EMD) system and of the Einstein-Born-Infeld (EBI) system. They have remarkable properties which are not seen for the corresponding solutions in the EBI and the EMD system.
We construct static axially symmetric solutions of SU(2) Einstein-Yang-Mills-dilaton theory. Like their spherically symmetric counterparts, these solutions are nonsingular and asymptotically flat. The solutions are characterized by the winding number n and the node number k of the gauge field functions. For fixed n with increasing k the solutions tend to ``extremal'' Einstein-Maxwell-dilaton black holes with n units of magnetic charge.
We construct a two-dimensional action that is an extension of spherically symmetric Einstein-Lanczos-Lovelock gravity. The action contains arbitrary functions of the areal radius and the norm squared of its gradient, but the field equations are second order and obey Birkhoff's theorem. In complete analogy with spherically symmetric Einstein-Lanczos-Lovelock gravity, the field equations admit the generalized Misner-Sharp mass as the first integral that determines the form of the vacuum solution. The arbitrary functions in the action allow for vacuum solutions that describe a larger class of int…
Aim of the paper is to obtain 2d analogue of the backreaction equation which will be useful to study final state of quantum perturbed spherically symmetric curved space times. Thus we take Einstein-massless-scalar $ψ$ tensor gravity model described on class of spherically symmetric curved space times. We rewrite the action functional in 2d analogue in terms of dimensionless dilaton-matter field $(χ=Φψ)$ where dilaton field $Φ$ is conformal factor of 2-sphere. Then we seek renormalized expectation value of quantum dilaton-matter field stress tensor operator by applying Hadamard rennormalization…
We solve the dilaton field equation in the background of a spherically symmetric black hole in bosonic or heterotic string theory with curvature-squared corrections in arbitrary d spacetime dimensions. We then apply this result to obtain a spherically symmetric black hole solution with dilatonic charge and curvature-squared corrections in bosonic or heterotic string theory compactified on a torus. For this black hole we obtain its free energy, entropy, temperature, specific heat and mass.
Static spherically symmetric solutions of 4d Brans-Dicke theory include a set of naked singularity solutions. Dilatonic effects near the naked singularities result in either a shielding or an antishielding effect from intruding massive test particles. One result is that for a portion of the solution parameter space, no communication between the singularity and a distant observer is possible via massive particle exchanges. Kaluza-Klein gravity is considered as a special case.
We derive exact magnetically charged, static and spherically symmetric black hole solutions of the four-dimensional Einstein-Born-Infeld-dilaton gravity. These solutions are neither asymptotically flat nor (anti)-de Sitter. The properties of the solutions are discussed. It is shown that the black holes are stable against linear radial perturbations.
We numerically integrate the semiclassical equations of motion for spherically symmetric Einstein-Maxwell theory with a dilaton coupled scalar field and look for zero temperature configurations. The solution we find is studied in detail close to the horizon and comparison is made with the corresponding one in the minimally coupled case.
The uniqueness theorem for static, spherically symmetric, asymptotically flat, higher dimensional phantom black holes, with non-degenerate event horizon , being the solutions of Einstein phantom/dilaton Maxwell/anti-Maxwell gravity systems is considered. Conformal positive energy theorem and conformal transformations authorize the crucial tools for exploiting the boundary conditions and conformal flatness.
We solve the dilaton field equation in the background of a spherically symmetric black hole in type II superstring theory with $α'^3$ corrections in arbitrary $d$ spacetime dimensions. We then apply this result to obtain a spherically symmetric black hole solution with $α'^3$ corrections in superstring theory compactified on a torus, coupled to such dilaton. For this black hole we obtain its mass, entropy, temperature, specific heat, and free energy.
We discuss the new class of static axially symmetric black hole solutions obtained recently in Einstein-Yang-Mills and Einstein-Yang-Mills-dilaton theory. These black hole solutions are asymptotically flat and they possess a regular event horizon. The event horizon is almost spherically symmetric with a slight elongation along the symmetry axis. The energy density of the matter fields is angle-dependent at the horizon. The static axially symmetric black hole solutions satisfy a simple relation between mass, dilaton charge, entropy and temperature. The black hole solutions are characterized by …
We derive Bogomolny equations for an Einstein-Yang-Mills-dilaton-$σ$ model (EYMD-$σ$) on a static spacetime, showing that the Einstein equations are satisfied if and only if the associated (conformally scaled) three-metric is flat. These are precisely the static metrics for which super-covariantly constant spinors exist. We study some general properties of these equations and then consider the problem of obtaining axially symmetric solutions for the gauge group SU(2).
We study generic Einstein-Maxwell-Kalb-Ramond-dilaton actions, and derive conditions under which they give rise to static, spherically symmetric black hole solutions. We obtain new asymptotically flat and non-flat black hole solutions which are in general electrically and magnetically charged. They have positive definite and finite quasi-local masses. Existing non-rotating black hole solutions (including those appearing in low energy string theory) are recovered in special limits.
We show in detail that the entropic force of the static spherically symmetric spacetimes with unusual asymptotics can be calculated through the Verlinde's arguments. We introduce three different holographic screen candidates, which are first employed thoroughly by Myung and Kim [Phys. Rev. D 81, 105012 (2010)] for Schwarzschild black hole solutions, in order to identify the entropic force arising between a charged dilaton black hole and a test particle. The significance of the dilaton parameter on the entropic force is highlighted, and shown graphically.
The intermediate and late-time behaviour of a massive Dirac field in the background of static spherically symmetric black hole solutions is investigated. The intermediate asymptotic behaviour of a massive Dirac field depends on the mass parameter as well as the wave number of the mode, while the late time behaviour has a power law decay rate independent of both.
We study black hole solutions in the Einstein-Gauss-Bonnet gravity with the dilaton and a negative ``cosmological constant''. We derive the field equations for the static spherically symmetric ($k=1$) and hyperbolically symmetric ($k=-1$) spacetime in general $D$ dimensions. The system has some scaling symmetries which are used in our analysis of the solutions. We find exact solutions, i.e., regular AdS solution for $k=1$ and a massless black hole solution for $k=-1$. Nontrivial asymptotically AdS solutions are obtained numerically in D=4 -- 6 and 10 dimensional spacetimes. For spherically sym…
A generalized dilaton action is considered of which the standard dilaton black hole and spherically reduced gravity are particular cases. The Arnowitt-Deser-Misner (ADM) and the Bondi-Sachs (BS) mass are calculated. Special attention is paid to both the asymptotic conditions for the metric as well as for the reference space-time. For the latter one we suggest a modified expression thereby obtaining a new definition of energy. Depending on the parameters of the model the Hawking radiation behaves like a positive or negative power of the mass.
We invesigate analytically both the intermediate and late-time behaviour of the massive scalar field in the background of static spherically symmetric black hole solution in dilaton gravity with arbitrary coupling constant. The intermediate asymptotic behaviour of scalar field depends on the field's parameter mass as well as the multiple number l. On its turn, the late-time behaviour has the power law decay rate independent on coupling constant in the theory under consideration.
We present the topological solutions of Einstein-dilaton gravity in the presence of a non-Abelian Yang-Mills field. In 4 dimensions, we consider the $So(3)$ and $So(2,1)$ semisimple group as the Yang-Mills gauge group, and introduce the black hole solutions with spherical and hyperbolic horizons, respectively. The solution in the absence of dilaton potential is asymptotically flat and exists only with spherical horizon. Contrary to the non-extreme Reissner-Nordstrom black hole, which has two horizons with a timelike and avoidable singularity, here the solution may present a black hole with a n…
The coupling of a dilaton to the $SU(2)$-Yang-Mills field leads to interesting non-perturbative static spherically symmetric solutions which are studied by mixed analitical and numerical methods. In the abelian sector of the theory there are finite-energy magnetic and electric monopole solutions which saturate the Bogomol'nyi bound. In the nonabelian sector there exist a countable family of globally regular solutions which are purely magnetic but have zero Yang-Mills magnetic charge. Their discrete spectrum of energies is bounded from above by the energy of the abelian magnetic monopole with u…
We numerically reanalyze static and spherically symmetric black hole solutions in an Einstein-Maxwell-dilaton system with a dilaton potential $m_{d}^{2}ϕ^{2}$. We investigate thermodynamic properties for various dilaton coupling constants and find that thermodynamic properties change at a critical dilaton mass $m_{d,crit}$. For $m_{d}\geq m_{d,crit}$, the black hole becomes an extreme solution for a nonzero horizon radius $r_{h,ex}$ as the Reissner-Nordström black hole. However, if $m_{d}$ is nearly equal to $m_{d,crit}$, there appears a solution of smaller horizon radius than $r_{h,ex}$. For …
Static, spherically symmetric monopole solutions of a spontaneously broken SU(2) gauge theory coupled to a massive dilaton field are studied in detail in function of the dilaton coupling strength and of the dilaton mass.
It is known that certain properties of charged dilaton black holes depend on a free parameter $β$ which controls the strength of the coupling of the dilaton to the Maxwell field. We obtain the energy associated with static spherically symmetric charged dilaton black holes for arbitrary value of the coupling parameter and find that the energy distribution depends on the value of $β$. With increasing radial distance, the energy in a sphere increases for $β= 0$ as well as for $β< 1$, decreases for $β> 1$, and remains constant for $β= 1$. However, the total energy turns out to be the same for all …
A numerical study of static, spherically symmetric sphaleron solutions in the standard model coupled to the dilaton field is presented. We show that sphaleron is surrounded by strong dilaton cloud which vanishes inside the sphaleron.
We compute the classical effective action of color charges moving along worldlines by integrating out the Yang-Mills gauge field to next-to-leading order in the coupling. An adapted version of the Bern-Carrasco-Johansson (BCJ) double-copy construction known from quantum scattering amplitudes is then applied to the Feynman integrands, yielding the prediction for the classical effective action of point masses in dilaton gravity. We check the validity of the result by independently constructing the effective action in dilaton gravity employing field redefinitions and gauge choices that greatly si…
We obtain the general static, spherically symmetric solution for the Einstein-Maxwell-dilaton system in four dimensions with a phantom coupling for the dilaton and/or the Maxwell field. This leads to new classes of black hole solutions, with single or multiple horizons. Using the geodesic equations, we analyse the corresponding Penrose diagrams revealing, in some cases, new causal structures.
We study spherically symmetric, asymptotically flat black hole solutions in the low-energy effective heterotic string theory, which is the Einstein gravity with Gauss-Bonnet term and the dilaton, in various dimensions. We derive the field equations for suitable ansatz for general D dimensions and construct black hole solutions of various masses numerically in D=4,5,6 and 10 dimensional spacetime with (D-2)-dimensional hypersurface with positive constant curvature. A detailed comparison with the non-dilatonic solutions is made. We also examine the thermodynamic properties of the solutions. It i…
We construct and study general static, spherically symmetric, magnetically charged solutions in Einstein-Maxwell-dilaton gravity in four dimensions. That is, taking Einstein gravity coupled to a ${\rm U}(1)$ gauge field and a massless dilaton$\unicode{x2014}$e.g., the action in the low-energy limit of string theory or Kaluza-Klein reduction$\unicode{x2014}$with arbitrary dilaton coupling, we build a three-parameter family of objects characterized by their mass, charge, and dilaton flux, generalizing the well known Garfinkle-Horowitz-Strominger black hole. We analyze the near-extremal and near-…
Alternatives to Einstein's theory of general relativity can be distinguished by measuring the parametrised post Newtonian parameters. Two such parameters $β$ and $γ$, equal to one in Einstein theory, can be obtained from static spherically symmetric solutions. For the graviton-dilaton system, as in Brans-Dicke or low energy string theory, we find that if $γ\ne 1$ for a charge neutral point star, then there exist naked singularities. Thus, if $γ$ is measured to be different from one, then it cannot be explained by these theories, without implying naked singularities. We also couple a cosmologic…
A numerical study of coupled to the dilaton field, static, spherically symmetric monopole solutions inspired by the Kaluza-Klein theory with large extra dimensions are presented. The generalized Prasad-Sommerfield solution is obtained. We show that monopole may have also the dilaton cloud configurations.
We study the coupled Einstein-Yang-Mills-Dilaton (EYMD) equations for a Fried\-mann-Le\-mai\-tre universe with constant curvature $k=1$. Our detailed analysis is restricted to the case where the dilaton potential and the cosmological constant vanish. Also assuming a static gauge field, we present analytical and numerical results on the behavior of solutions of the EYMD equations. For different values of the dilaton coupling constant we analyze the phase portrait for the time evolution of the dilaton field and give the behavior of the scale factor. It turns out that there are no inflationary st…
The quantum mechanics of black holes in generic 2-D dilaton gravity is considered. The Hamiltonian surface terms are derived for boundary conditions corresponding to an eternal black hole with slices on the interior ending on the horizon bifurcation point. The quantum Dirac constraints are solved exactly for these boundary conditions to yield physical eigenstates of the energy operator. The solutions are obtained in terms of geometrical phase space variables that were originally used by Cangemi, Jackiw and Zwiebach in the context of string inspired dilaton gravity. The spectrum is continuous i…
Chamorro and Virbhadra studied, using the energy-momentum complex of Einstein, the energy distribution associated with static spherically symmetric charged dilaton black holes for an arbitrary value of the coupling parameter $γ$ which controls the strength of the dilaton to the Maxwell field. We study the same in Tolman's prescription and get the same result as obtained by Chamorro and Virbhadra. The energy distribution of charged dilaton black holes depends on the value of $γ$ and the total energy is independent of this parameter.
We reconsider here the problem of finding the general 4D spherically symmetric, asymptotically flat and time-independent solutions to the lowest-order string equations in the $\ap$ expansion. Our construction includes earlier work, but differs from it in three ways. (1) We work with general background metric, dilaton, axion and $U(1)$ gauge fields. (2) Much of the original solutions were required to be nonsingular at the apparent horizon, motivated by an interest in finding string corrections to black hole spacetimes. We relax this condition throughout, motivated by the realization that string…
We construct sequences of axially symmetric multisphaleron solutions in SU(2) Yang-Mills-dilaton theory. The sequences are labelled by a winding number $n>1$. For $n=1$ the known sequence of spherically symmetric sphaleron solutions is obtained. The solutions within each sequence are labelled by the number of nodes $k$ of the gauge field functions. The limiting solutions of the sequences correspond to abelian magnetic monopoles with $n$ units of charge and energy $E \propto n$.
The properties of static spherically symmetric black holes, which are either electrically or magnetically charged, and which are coupled to the dilaton in the presence of a cosmological constant, are considered. It is shown that such solutions do not exist if the cosmological constant is positive (in arbitrary spacetime dimension >= 4). However, asymptotically anti-de Sitter black hole solutions with a single horizon do exist if the cosmological constant is negative. These solutions are studied numerically in four dimensions and the thermodynamic properties of the solutions are derived. The ex…
We study the propagation of classical electromagnetic waves on the simplest four-dimensional spherically symmetric metric with a dilaton background field. Solutions to the relevant equations are obtained perturbatively in a parameter which measures the strength of the dilaton field (hence parameterizes the departure from Schwarzschild geometry). The loss of energy from outgoing modes is estimated as a back-scattering process against the dilaton background, which would affect the luminosity of stars with a dilaton field. The radiation emitted by a freely falling point-like source on such a back…
We analyze the stability of the geodesic curves in the geometry of the Gibbons-Maeda-Garfinkle-Horowitz-Strominger black hole, describing the space time of a charged black hole in the low energy limit of the string theory. The stability analysis is performed by using both the linear (Lyapunov) stability method, as well as the notion of Jacobi stability, based on the Kosambi-Cartan-Chern theory. Brief reviews of the two stability methods are also presented. After obtaining the geodesic equations in spherical symmetry, we reformulate them as a two-dimensional dynamic system. The Jacobi stability…
Classical solutions of the Yang-Mills-dilaton theory in Euclidean space-time are investigated. Our analytical and numerical results imply existence of infinite number of branches of dyonic type solutions labelled by the number of nodes of gauge field amplitude $W$. We find that the branches of solutions exist in finite region of parameter space and discuss this issue in detail in different dilaton field normalization.
Static spherically symmetric black hole solution is obtained in the framework of Einstein-dilaton theory with nonlinear Maxwell and Yang-Mills fields of power-law type. It is observed that black hole might have two horizons similarly as it takes place for linear gauge fields case. Thermodynamics of the black hole is studied, namely the behaviour of temperature is examined and the first law is written. The concept of extended phase space is also utilized, namely we have written and analyzed the equation of state for the black hole and examined the Gibbs free energy. The Gibbs free energy shows …
