Slowly rotating Einstein–Maxwell-dilaton black hole and some aspects of its thermodynamics
A slowly rotating black hole solution in Einstein–Maxwell-dilaton gravity is considered. Using the obtained solution we investigate thermodynamic functions such as the black hole’s temperature, entropy and heat capacity. In addition, to examine the thermodynamic properties of the black hole an extended technique is applied. The equation of state of Van der Waals type is obtained and investigated. It has been shown that the given system has phase transitions of the first and of the zeroth order below a critical temperature, which is a notable feature of the black hole. A coexistence relation for two phases is also considered and the latent heat is calculated. In the end, critical exponents are calculated.
The thermodynamics of black holes has been investigated intensively for over four decades. This area of research is considered to bring to light some links between general relativity and quantum mechanics and might be very useful for the verification of different approaches to quantum gravity . It is supposed that all the approaches in the quasiclassical limit should lead to the same thermodynamic behaviour of black holes and to be in agreement with the laws of black hole mechanics  which were developed without any knowledge about the underlying theory of quantum gravity. It might be treated as a theory that allows one to correct or even get rid of some deficit approaches to quantum gravity. But it should be noted that such different approaches to quantum gravity as String Theory  and Loop Quantum Gravity  give rise to the same expression for the black hole entropy–area relation given by the celebrated Bekenstein–Hawking formula.
New great interest in thermodynamics of black holes has arisen in the past decade. This revival of activity can be explained by the fact that in recent work one proposed to extend the complex of thermodynamic variables which had been used in black holes’ theory and consider the cosmological constant as one of those thermodynamic values [5, 6, 7, 8, 9, 10]. It was suggested that for a gravitational system for example for a black hole such a standard notion of thermodynamics as pressure can be introduced, being proportional to the cosmological constant. In the case of a Reissner–Nordstrom–AdS (RN-AdS) black hole one has a simple relation between the thermodynamic pressure and the cosmological constant: \(P=-\Lambda /8\pi \) [5, 6, 7, 8]. Since the cosmological constant here is supposed to be a variable it changes the identification of the black hole mass in thermodynamics. In contrast to the well-established treatment of the mass as the black hole’s internal energy, in the extended approach it is considered to be like the thermodynamic enthalpy, namely: \(M=H=U+PV\). The second point that is strongly related to the introduced notion of the pressure is the definition of its thermodynamically conjugate value, namely it is the volume which can be introduced by the derivative, \(V=(\partial H/\partial P)_S\). The extended (enlarged) thermodynamic phase space allowed one to examine the phase behaviour of black holes, namely it was shown that a charged black hole possesses a phase transition that is completely analogous to Van der Waals liquid–gas phase transition and below the critical point it has a phase transition of the first order . It should be noted that a richer thermodynamic behaviour appears in the case of a more general setting, namely for Born–Infeld theory or for black holes in a space of higher dimensions. For example, a reentrant phase transition appears [10, 11] or a tricritical (triple) point was observed [12, 13]. The facts mentioned stimulated deep interest and prolific investigations in this area of research which obtained the special name: black hole chemistry, and this name reflects the similarity in thermodynamic behaviour of black holes and ordinary condensed matter systems such as gases, liquids, multicomponent fluid systems. An overview of recent developments and possible future directions of this subfield of black hole thermodynamics is given in .
General Relativity is a very successful theory in the explanation of various phenomena from planetary up to cosmological scales. But nonetheless, different approaches to quantum gravity, namely String Theory, are supposed to modify the gravitational Einstein term replacing it with a sum of ones which takes into account higher orders corrections of curvature. In the low energy limit of the String Theory the standard Einstein term plus a nonminimally coupled dilaton field can be obtained . It is worth stressing that dilaton field minimally coupled to Einstein’s gravity appears as a result of a dimensional reduction of the standard General Relativity in higher dimensions onto a space of lower dimension. Thus, such a low energy limit of the String Theory can be considered as Einstein gravity with an additional dilaton field. This model has attracted considerable attention in the past two decades, namely different types of black holes/strings solutions were investigated [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35]. We point out that in most of this work static solutions were studied, whereas the rotating solutions were examined for some particular choice of the dilaton coupling parameter [36, 37, 38, 39, 40, 41, 42, 43, 44]. It should be noted that special interest has been given to the investigation of Einstein-dilaton black holes with an additional dilaton potential of a so called Liouville-type form [21, 22, 24, 27, 29, 45, 46, 47, 48, 49, 50, 51]. The Liouville-type potential allows one to introduce terms which can be treated as a generalization of the known cosmological constant term. This fact can be used for examination of possible extensions of the AdS–CFT correspondence. It was also supposed that linear dilaton spacetime, which appears as a near horizon limit of a dilatonic black hole, might possess some holographic properties . On the other side new black hole solutions which are not asymptotically flat nor of AdS-type (or dS-type) might be a good laboratory for application or testing of methods developed for the above-mentioned types of black hole solutions. The interest in the dilatonic black holes with Liouville dilaton potential was quite great; namely, black hole solutions with different types of nonlinear electromagnetic field coupled to a dilaton field were obtained [53, 54, 55, 56, 57, 58]. Considerable attention was paid to the investigation of thermodynamic properties of the corresponding dilatonic black holes. It should be noted that the standard approach and the extended technique were applied for the study of their thermodynamics [47, 54, 57, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69].
In our work we examine a slowly rotating Einstein–Maxwell-dilaton black hole with a linear Maxwell field and electromagnetic–dilaton coupling. We also take into account a dilaton potential of the Liouville form which consists of two terms; in particular one of these is supposed to be a kind of generalization of a cosmological constant term. We also consider the thermodynamics of given black hole solution. For the first, assuming that the corresponding cosmological constant is fixed we obtain temperature and heat capacity. Secondly, using the modern technique where the thermodynamic pressure is introduced we obtain the equation of state and derive the Gibbs potential and investigate the critical behaviour.
This work is organized in the following way. In Sect. 2 the slowly rotating black hole is obtained and examined. Section 3 is devoted to the thermodynamics of the black hole; relations for the temperature and heat capacity are investigated. In Sect. 4 the equation of state and the Gibbs potential for the black hole are obtained and investigated. Section 5 is devoted to the derivation of critical exponents. Finally, Sect. 6 contains some conclusions.
2 Field equations and their solution
3 Thermodynamics of the black hole
The further increase of the absolute value of \(\Lambda \) leads to a decrease of the peak of the heat capacity and its subsequent disappearance. Thus, we can conclude that when the absolute value of \(\Lambda \) is large enough the heat capacity \(C_Q\) is positive for all values of \(r_+\) greater than the boundary value mentioned above when the temperature T (28) equals zero; as a result the system is thermodynamically stable for all these values of \(r_+\).
4 Extended thermodynamics
For a long time it has been supposed that black hole thermodynamics should be considered in a “fixed” background, which means that for the theories with cosmological constant, namely for AdS-type black holes, the cosmological constant is held fixed. An extension of standard thermodynamic phase space has been proposed recently. It was supposed that the cosmological constant might be varied . It leads to interesting physical consequences and implications. As emphasized earlier, the extended thermodynamic phase space might bring about a better understanding of the black hole thermodynamics from a broader point of view and it allows one to reveal new ties with the thermodynamics of real systems, namely liquid–gas systems which are described by the Van der Waals equation of state [9, 10]. It was also noted that the variation of the cosmological constant allows one to solve some important problems in black hole thermodynamics, namely in the extended phase space the Smarr relation can be recovered. It was also pointed out that the extended thermodynamics which introduces the notion of a thermodynamic volume of a black hole satisfies the inverse isoperimetric inequality .
Figure 9 represents the two types of possible changes. The left graph shows the dependence for fixed value of n and for several different values of the parameter \(\alpha \). One can see that for smaller values of \(\alpha \) the \(L=L(T)\) dependence is nonmonotonic with some specific maximum point and then decreasing to zero. Increasing of the parameter \(\alpha \) gives rise to the shift of the maximum point to lower temperatures. At the same time increasing of dimension n when \(\alpha \) is fixed leads to increase of the absolute value of the latent heat L with corresponding shift of the maximum point to higher temperatures. It should be pointed out that for the domain where we have the phase transition of the zeroth order it is not possible to calculate the latent heat with the help of Eq. (60), because this formula is obtained under the assumption that the Gibbs potential is continuous, which does not hold for the phase transition of the zeroth order. But reaching the critical temperature the phase transition transforms into a phase transition of the second order where the latent heat is equal to zero, so any of the represented curves should be continued by a sort of other curve reaching zero latent heat at the critical temperature corresponding to the very same parameters as n, \(\alpha \), b and q, which represent the given curve.
5 Critical exponents
Since the system possesses a critical point it is interesting to investigate the behaviour of the system in the domain close to the critical point. In general we follow the approach developed in Refs. [9, 10].
In our work we have considered a slowly rotating black hole in the framework of Einstein–Maxwell-dilaton theory. In this theory an additional dilaton potential term of the so-called Liouville form is taken into account and this form allows us to obtain a cosmological constant term. A similar problem was examined in [48, 50] but in the first reference of these the cosmological constant term is not taken into account, while in the second work this term takes a slightly different form. We have obtained the solution of the corresponding Einstein and field equations, which is in a perfect agreement with the results of the work of [47, 48, 50]. In the limit when the dilaton parameter goes to zero the slowly rotating Kerr–Newmann solution is recovered . It should be pointed out that the solution obtained here can be treated as a generalization of the corresponding solution in . We note that the obtained solution possesses a complicated causal structure, which in general respects is analogous to a nonrotating dilaton black hole .
We have investigated the black hole thermodynamics making the assumption that the cosmological constant is held fixed. Firstly, it was shown that, for negative cosmological constant, the temperature shows a nonmonotonous behaviour as a function of the horizon radius \(r_+\). It is shown that for small radius \(r_+\) the temperature (28) becomes negative and it gives rise to the conclusion that a black hole with such a small radius would be unstable. It is also demonstrated that the point when the temperature becomes equal to zero is the very same point as when the heat capacity \(C_Q\) changes sign from positive to negative, and this fact confirms the thermodynamic instability of the black hole. The heat capacity was shown to have two points of discontinuity which separate stable and unstable domains. But for cosmological constant large enough in absolute value the points of discontinuity merge with each other and finally disappear, transforming into a high peak, which diminishes with the following increasing of the cosmological constant (Fig. 5). It means that when the cosmological constant is large in absolute value we have a black hole which might be stable for an arbitrarily small radius of the horizon, which peculiarity, to the best of our knowledge, has not been paid attention to in previous work.
We also examined the thermodynamics using the extended technique. It was shown that the obtained black hole solution allows one to obtain a equation of state of Van der Waals type; see Eq. (39) (or (42)). Similar to recently obtained results of  our system possesses a domain of the first as well as of the zeroth order phase transitions. The appearance of the zeroth order phase transition is directly related to the existence of dilaton-Maxwell fields’ coupling, which is described by the parameter \(\alpha \). The main conclusion we should point out here is the fact that with increasing of the coupling the domain where the zeroth order phase transition takes place becomes wider. For small enough \(\alpha \) this domain, where a zeroth order phase transition happens, is negligibly small and it is hardly visible on the graph (see the left graph of Fig. 6), but when the coupling parameter \(\alpha \) increases this domain drastically increases and becomes notable on the graph. We also note that the zeroth order phase transition takes place also for other systems with dilaton–electromagnetic coupling with a different type of electromagnetic field action; see , but in that work it was pointed out that the discontinuity of the Gibbs potential which gives rise to the zeroth order phase transition is related not only to the dilaton–electromagnetic field coupling constant \(\alpha \). In our case we can state that the existence of the zeroth order phase transition is completely caused by the coupling between the fields. In the case this coupling disappears and the zeroth order transition would not occur. Using the equation of state and Maxwell’s equal area law we also obtained the coexistence relation for the system in the domain where the first order phase transition takes place. Having used the Clapeyron equation the latent heat was calculated numerically, and we note that these calculations are valid on the very same domain. Finally, we calculated the critical exponents. They are shown to be the same as for an Einstein–Maxwell black hole.
This work was partly supported by Project FF-30F (No. 0116U001539) from the Ministry of Education and Science of Ukraine.
- 15.M.B. Green, J.H. Schwartz, E. Witten, Superstring Theory, CUP (1987)Google Scholar
- 76.Z. Dayyani, A. Sheykhi, M.H. Dehghani, S. Hajkhalili. arXiv:1709.06875
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