Charged anti-de Sitter scalar-tensor black holes and their thermodynamic phase structure
In the present paper we numerically construct new charged anti-de Sitter black holes coupled to nonlinear Born-Infeld electrodynamics within a certain class of scalar-tensor theories. The properties of the solutions are investigated both numerically and analytically. We also study the thermodynamics of the black holes in the canonical ensemble. For large values of the Born-Infeld parameter and for a certain interval of the charge values we find the existence of a first-order phase transition between small and very large black holes. An unexpected result is that for a certain small charge subinterval two phase transitions have been observed, one of zeroth and one of first order. It is important to note that such phase transitions are also observed for pure Einstein-Born-Infeld-AdS black holes.
PACS: 04.50.Kd; 04.70.Bw; 04.25.D
In the last decade anti-de Sitter (AdS) black holes and especially their thermodynamics have attracted considerable interest due to the AdS/CFT duality. According to the AdS/CFT conjecture the thermodynamics of the AdS black holes is related to the thermodynamics of the dual conformal field theory (CFT) residing on the boundary of the AdS space [1, 2]. In their pioneering work Hawking and Page  showed the existence of a phase transition between the AdS black hole and thermal AdS space. The AdS/CFT duality provides us with a new tool to study the phase transitions in the dual CFT theories on the base of studying the phase transitions of the AdS black holes.
Charged AdS black holes and their thermodynamics have been studied not only within the linear Maxwell electrodynamics but also within the framework of the nonlinear electrodynamics of Born-Infeld. This nonlinear electrodynamics was first introduced by Born and Infeld in 1934 as an attempt to obtain a finite energy density model for the electron . The interest for nonlinear electrodynamics has been later revived in the context of string theory. It arises naturally in open strings and D-branes  – . Nonlinear electrodynamics models coupled to gravity have been discussed in different aspects (see, for example,  –  and references therein). In particular, black holes coupled to Born-Infeld electrodynamics, including in the presence of a cosmological constant, have been studied in numerous papers , – , and their thermodynamics in  – .
The aim of the present paper is to study the charged AdS black holes coupled to nonlinear Born-Infeld electrodynamics and their thermodynamics within the framework of a certain class of scalar-tensor theories (STT). Scalar-tensor theories of gravity are among the most natural generalizations of general relativity (GR) and arise in string theory and higher dimensional gravity theories . The fundamental question is whether objects found and studied in the frame of GR, such as black holes, would have different properties in the frame of the scalar-tensor theories. Recent studies on asymptotically flat, scalar-tensor black holes coupled to nonlinear electrodynamics show that the presence of a scalar field in the gravitational sector leads to interesting and serious consequences for the black holes. In the general case the scalar field restricts the possible causal structures in comparison to the pure Einstein gravity [38, 39]. For some classes of scalar-tensor theories  there even exist non-unique scalar-tensor black hole solutions with the same conserved asymptotic charges. In other words the spectrum of asymptotically flat black hole solutions coupled to nonlinear electrodynamics in the STT is much richer and much more complicated than in general relativity. This shows that the scalar-tensor black holes coupled to nonlinear electrodynamics and other sources with non-vanishing trace of the energy-momentum tensor (in four dimensions) constitute an area that deserves further study. In view of the current great interest in the AdS black holes, the natural next step is to study the scalar-tensor black holes coupled to nonlinear electrodynamics in spacetimes with AdS asymptotic.
In this paper we construct new numerical solutions describing charged AdS scalar-tensor black holes coupled to nonlinear Born-Infeld electrodynamics and we make a parameter study of their properties. The thermodynamics of the constructed solutions in canonical ensemble is also studied. We found that for large values of the Born-Infeld parameter and for a certain interval of intermediate values of the charge there exist a first-order phase transition between small and very large black holes. An interesting result is that for a small subinterval one more phase transition exists which is of zeroth order since the thermodynamic potential is discontinuous there – it jumps to a lower value. Such phase transition of first and zeroth order can be observed also in pure Einstein gravity, i.e. when no scalar field is present.
The phase structure of the Einstein-Born-Infeld black holes and the possible Hawking-Page transitions have been examined in  and . Hawking-Page phase transitions have been studied for different black holes, for example, Einstein-Maxwell black holes in spacetime with different dimensions [48, 49], in three-dimensional spacetime [50, 51], in higher-derivative gravity [52, 53], in higher-curvature gravity , and in unusual electrodynamics which does not restore the Maxwell electrodynamics in the weak-field limit . Phase transitions between solitons and black holes in asymptotically AdS/ spaces have been studied in .
2 Formulation of the problem
The most general form of the action in scalar-tensor theories is
where is the bare gravitational constant and is the Ricci scalar curvature with respect to the spacetime metric . , , and are functions of the scalar field and their specific choice determines the scalar-tensor theory completely. In order for the gravitons to carry positive energy the function must be positive (), while the non-negativity of the scalar field energy requires that . The action of the sources is and there is no direct coupling between the sources of gravity and the scalar field in order for the weak equivalence principal to be satisfied.
The action (1) is in the so called Jordan frame which is the physical frame but it is more convenient to work in the Einstein frame. The relations between the two frames are
where and are correspondingly the metric and the scalar field in the Einstein frame. The additional introduction of the functions
leads to the following action in the Einstein frame:
where is the Ricci scalar curvature with respect to the spacetime metric . In the Einstein frame there is direct coupling between the sources of gravity and the scalar fields through the coupling function and the specific choice of the scalar-tensor theory is completely determined by this function and by the potential of the scalar field .
Here we will consider nonlinear electrodynamics and its action in the Einstein frame is given by
where is the Lagrangian of the nonlinear electrodynamics. The equations defining the functions and are
where is the electromagnetic field strength tensor and stands for the Hodge dual with respect to the metric .
The Lagrangian of the Born-Infeld nonlinear electrodynamics is
where is a parameter and in the limit the linear Maxwell electrodynamics is restored.
The variation of the action (6) leads to the following field equations
In the present paper we are interested in spacetimes with anti-de Sitter asymptotic structure and thus we will consider the following type of potential:
where is a constant. The coefficient is a standard normalizing factor.
Let us explain the reasons for choosing the potential (13). For this purpose we consider the Jordan frame and we require the metric to have AdS asymptotic111We mean AdS asymptotic corresponding to a cosmological term . in this frame. Then we have two possibilities – either the factor is finite at infinity (i.e. ) or it is divergent at infinity. Although the case when is divergent at infinity could be of some interest, it is not generic and in fact is degenerate since it corresponds to zero effective gravitational constant at infinity. Here we consider only ”physically well-behaved” solutions with finite222Throughout this paper, without loss of generality, we set . . Using the (Jordan frame) field equations and especially the equation for , one can show that the finiteness of at infinity requires
The simplest (Jordan frame) potential satisfying this condition333More precisely, we mean the simplest potential in the framework of the class of scalar-tensor theories considered in the present paper. For this class of scalar-tensor theories the denominator in (2) is finite. , i.e. allowing simultaneously AdS asymptotic for the metric and finite , is which corresponds exactly to the Einstein frame potential (13) for . The conditions (2) written in the Einstein frame take the form
From here it is obvious that is indeed the simplest choice of a potential admitting the desired properties.
One can show, as we shall see below, that for the scalar-tensor theories under consideration, the asymptotic behavior of the factor is and it guarantees the same anti-de Sitter asymptotic structure simultaneously in both the Jordan and the Einstein frames. Moreover, the asymptotic behavior of guarantees that the mass of the Jordan frame solution is the same as that of the Einstein frame solution. This can be easily seen from the results in Appendix B.
Another reason for considering finite comes from the thermodynamics. The black hole thermodynamics in scalar-tensor theories is naturally defined in the Einstein frame as discussed in Appendix A. When is divergent at infinity the conformal transformation does not preserve the AdS asymptotic when we move from the Jordan to the Einstein frame. In this case the expected thermodynamic would be rather different from that of the AdS black holes if it could be defined at all because of the rather unusual spacetime asymptotic.
3 Basic equations
3.1 The reduced system
Here and below we will work in a system of units in which , where is the magnetic permeability of the vacuum. In this system
We will consider a static and spherically symmetric spacetime and the ansatz for the metric (in the Einstein frame) is
An important property of the Born-Infeld electrodynamics is the electric-magnetic duality symmetry of the theory  – . It is sufficient to study only the magnetically charged case, and the electrically charged solution can be obtained from the magnetically charged solution using the electric-magnetic rotation defined by
where the functions denoted by correspond to the dual solution, is the magnetic charge, is the charge of the dual solution, and
In the magnetically charged case the electromagnetic field strength tensor is
where is the magnetic charge. For the functions and we obtain
The truncated Born-Infeld Lagrangian is
Using the metric (16) the field equations reduce to the following system of coupled ordinary differential equations
These are four equations for only three unknown functions , and but the self-consistency of the system is guaranteed by the Bianchi identity.
In pure Einstein theory the solution describing Schwarzschild-AdS black holes is
where is the mass of the black hole. The asymptotic of the metric function at infinity in our problem is the same and this means that is unbounded at infinity. From a numerical point of view it is convenient to introduce a new unknown function (which is finite at infinity in the considered class of theories) using the substitution
The first equation (29) is decoupled and it can be solved separately once a solution for the functions and is obtained.
We will consider a class of scalar-tensor theories for which 444Constant corresponds to Brans-Dicke scalar-tensor theory.. The investigation of the case with is similar. Although we restrict ourselves to the obtained results are qualitatively555As the numerical results show the picture is not only qualitatively but also quantitatively close for all scalar-tensor theories with the same . the same even for more general coupling functions for which .
In the chosen ansatz for the metric (16) the temperature is given by the following relation
3.2 Asymptotic behavior
The asymptotic behavior of the functions , and at is
where is a constant and is the mass of the black hole in the Einstein frame (for the definition of the mass in the AdS spaces see Appendix A). Let us compare the asymptotic behavior of these functions in AdS and in asymptotically flat spacetime. In AdS spacetime the scalar field decreases as and this is much faster than in the asymptotically flat case where the scalar field decreases as . The function also decreases much faster in AdS spacetime, where the leading term is proportional to compared to the leading term in the asymptotically flat spacetime. The asymptotic behavior of up to first order in is the same.
3.3 Qualitative properties
We can obtain some information for the black hole solutions using equations (25) and (29) and the boundary conditions. We prove that the black holes under consideration have simpler causal structure than the black holes in pure Einstein-Born-Infeld theory in AdS spacetime and we also obtained some information for the behavior of the unknown functions. The general properties of the considered scalar-tensor black holes coupled to Born-Infeld nonlinear electrodynamics in AdS spacetime can be summarized as follows:
The black holes have only one event horizon and extremal black holes do not exist.
The scalar field is negative on the event horizon, it has no zeros and increases monotonically.
The metric function is positive on the event horizon, it has no zeros and decreases monotonically.
To prove these statements we should first note that it can be easily shown that for the Born-Infeld electrodynamics the following inequality is satisfied:
We will first prove that the considered black holes have only one horizon. Let us assume that they have two horizons and , where . When we integrate equation (25) in the interval we obtain
The horizons are defined as the points where so the left-hand side of the equation is zero. But from (36) and the fact that we consider it is obvious that the right-hand side is strictly positive. So we reached a contradiction and this means that two black hole horizons cannot exist.
In order to prove that an extremal black hole cannot exist we will examine again equation (25), that is
Let us assume that we have a degenerate event horizon at . The right-hand side of the equation is strictly positive for every , 666We are not interested in the case because this corresponds to naked singularity. but the left-hand side is zero and we reach a contradiction.
We can investigate the behavior of the function in the domain ( is the radius of the event horizon) using equation (25). As we already noted the right-hand side of (25) is positive. Integration of equation (25) from to gives
This shows that in the domain . Therefore increases monotonically. Since the asymptotic value of the scalar field at infinity is zero we can conclude that should be negative on the horizon and monotonically increasing to zero. From (38) taking the limit we find that .
Finally, decreases monotonically since the right-hand side of equation (29) is always negative. The asymptotic value of at infinity is zero so we can conclude that it is positive on the event horizon and monotonically decreasing to zero.
All of the above presented propositions remain valid for different nonlinear electrodynamics for which the relation (36) holds.
3.4 Dimensionless quantities
Therefore, one may generate in this way a family of new solutions. Thus, the mass, the temperature and the entropy of the new solutions are given by the formulas
As a consequence of the rigid symmetry we may restrict our study to the case by introducing the dimensionless quantities
where the choice of requires
3.5 Posing the Boundary-Value Problem
The domain of integration is half-infinite where the horizon, which is the left boundary, is a priori unknown. We impose the following boundary conditions:
— At infinity:
where is the mass of the black hole in the Einstein frame (see Appendix A);
— On the horizon:
The following regularization condition should also be fulfilled on the horizon:
In this way the governing equations (29)–(31) together with the above boundary conditions (43)–(46), complement a well-posed boundary value problem. Since the position of the event horizon (the left boundary of the integration interval) is not fixed, moreover it is unknown777Such BVPs are known in mathematical physics as BVPs of the Stefan kind. and there is no analytic way to determine it, we can use one of these boundary conditions to determine it. We can introduce a new shifted independent variable . In this way the domain of integration is completely defined – and the radius of the horizon will appear explicitly in the equations as an unknown parameter. So, we have one parametric problem, which can be considered as a spectral-like nonlinear BVP.
Let us emphasize that there is an alternative formulation of the problem where the horizon is an input parameter. In this case we do not need a shifting map and the mass becomes the sought parameter. We used this method of solution for justification of our results.
3.6 Method of solution
The system of ordinary differential equations (29)–(31) is coupled and nonlinear. It does not admit a global analytic solution and only some local expansions are possible. This is the reason to solve it numerically by using the reliable Continuous Analog of Newton’s Method, which is distinguished by its quadratic convergence in the vicinity of a localized root [62, 63]888Let us note that in our case the roots appear to be functions in Banach space.. Yet, the method has been already successfully applied to similar problems (see, for example, [64, 38, 39, 46]).
4 Discussion of numerical results
As we noted in the previous sections we will consider constant parameter ( is defined by equation (12)). The results do not change qualitatively for different and we will present the results for . 999This value is consistent with the current observational data .
In Figs. 1 and 2 the metric functions and and the scalar field for , , and for several values of the black hole mass are presented. An important justification of our results is that, up to the leading terms, the asymptotic form of the numerically obtained functions , , and at infinity is the same as what is predicted in (33), (34) and (35), respectively.
In Fig. 3 the horizon and the temperature as functions of the mass for sequences of black hole solutions are plotted for and . Fig. 4 represents two magnifications of the enclosed regions on the plot in Fig. 3. In Fig. 5 the value of the scalar field on the event horizon and the constant which occurs in the asymptotic expansion of the function at infinity (see equation (35)) as functions of the mass are plotted. In Figs. 6 and 7 the quantities , , and as functions of the charge are plotted.
When extremal black hole solutions exist in Einstein-Born-Infeld theory in AdS spacetime. In the presence of scalar field extremal black holes cannot exist as we have already proved. As we see from the plot for and (this combination satisfies ) in the right panel of Fig. 3, when decreases the temperature first reaches values close to zero and afterward increases. The reason is that the absolute value of the scalar field rises considerably in this region, as we can see in Fig. 5, and prevents the reaching of an extremal black hole. The same situation can be seen from the plot in Fig. 6 when we increase the magnetic charge (see, for example, the curve corresponding to ).
The numerical results are obviously subject to and possess the properties in Subsection 3.3. All of the dependences have similar qualitative behavior in the parameter space that we have studied. The only exception is the function because its qualitative behavior depends also on the value of the parameter (more details on this subject will be given in the next sections).
5 Thermodynamic phase structure
To study the thermodynamics101010Some clarifying comments on the black hole thermodynamics in scalar-tensor theories are given in Appendix A. of the black holes under consideration we need to find the conserved charges of our system first. Throughout the years many methods have been developed to calculate the conserved charges of the gravitational configurations. A related problem is the computation of the gravitational action of a noncompact spacetime. When evaluated on noncompact solutions, the bulk Einstein term and the boundary Gibbons-Hawking term are both divergent. The remedy is to consider these quantities relative to those associated with some background reference spacetime, whose boundary at infinity has the same induced metric as that of the original spacetime. This substraction procedure is however connected to difficulties: the choice of the reference background is by no means unique and it is not always possible to embed a boundary with a given induced metric into the reference background. A new method, free of the aforementioned difficulties, was proposed in . This method (called the counterterm method) consists of adding a (counter)term to the boundary at infinity, which is a functional only of the curvature invariants of the induced metric on the boundary. Unlike the substraction procedure, this method is intrinsic to the spacetime of interest and is unambiguous once the counterterm which cancels the divergences is specified. In the present work we use, namely, the counterterm method in order to compute the mass and gravitational action. The technical details are presented in Appendix B.
We will study the thermodynamic phase structure for black holes in the canonical ensemble where the magnetic charge is fixed. The associated thermodynamical potential in the case of the canonical ensemble, as one should expect from pure thermodynamical considerations, is the Helmholtz free energy . This can be proven with more rigor by calculating the Euclidean action as it is done in Appendix B .
Let us note that our definition of the free energy is intrinsic. Our computation based on the counterterm method makes no reference to any other solution of the field equations. This should be contrasted with the method used in  where the authors compute the action using the extremal solution (with fixed charge) as reference background. As we already showed in our case there are no extremal solutions and therefore the reference background method of  is unapplicable.
The thermodynamic phase structure depends on the value of magnetic charge and the parameter in the Born-Infeld Lagrangian. The numerical results show that for large values of the parameter the function has two inflection points defined by
These inflection points, which we will denote by and , separate three charge intervals given by , and . Below we will consider each of these intervals in detail.
For small values of the parameter the inflection points disappear and the phase structure is the same for all the charges. The numerical results show that for all of the studied values of the parameter (including the case which corresponds to pure Einstein gravity) the critical value of the parameter which separates the regions of small and large , is .
5.1 Phase structure for large values of the parameter
As we noted for large values of the parameter the function has two inflection points – and , which gives us three charge intervals defined by , and . Formally we call these three cases black hole with small, intermediate and large charge respectively. On its own side the intermediate charge interval has to be divided into three subintervals , and ( and are defined and discussed in details in the following subsections). As we show below, there are phase transitions for charges belonging to the second and the third subintervals.
All the results for the scalar-tensor black holes given in this subsection are computed for fixed values of the parameters and . For this choice of parameters the values of the critical charges are and . The value of is the same (within the numerical error) as in the pure Einstein theory (i.e. for ). The reason is that the corresponding inflection point, defined by equation (47), is situated in a region of the parameter space where the contribution of the scalar field is very small. The value of in the case of pure Einstein theory – , is a bit smaller than in the case when scalar field is present. The numerical results for larger confirm that increases slightly when we increase the value of the coupling constant .
The values of the charges and for the chosen values of the parameters and , are and and the corresponding values in pure Einstein theory are and . The numerical experiments show that the values of and increase slightly with the increase of .
The temperature as a function of the horizon radius is shown in Fig. 8 for the cases of small (for example =0.07), intermediate (for example =0.115), and large (for example =0.40) charges . The free energy as a function of the temperature for small and intermediate charges is shown in Fig. 9 and magnifications of specific regions are presented in Fig. 10. The function for various charges, including large charges, is given in the left panel of Fig. 12. In Figs. 11 and 12 (right panel) we also give the dependences and for Born-Infeld-AdS (BIAdS) black holes in the pure Einstein gravity for comparison. The figures show that the thermodynamic phase structures for the scalar-tensor BIAdS black holes and the Einstein BIAdS black holes are qualitatively the same except in the case of charges satisfying where additional phases of the scalar-tensor BIAdS black holes are present. Below we consider in detail the thermodynamic phase structure for the intervals of small, intermediate and large charges.
The phase structure and the phase transitions can be alternatively studied with the so-called off-shell formalism or, in other words, off-equilibrium formalism. It is thoroughly presented in  and applied, for example, in the papers on black-hole phase transitions by Myung and coauthors [50, 51, 47]. The advantage of that formalism is that it gives a simple and clear interpretation of the thermodynamic phases. In the canonical ensemble the different phases of the system, both stable and unstable, correspond to extrema of the so-called off-shell free energy. In that formalism, the origin of the branches in the diagram, that correspond to the different phases, can be also simply explained. The phase transitions have been studied with the application of the off-shell formalism in Appendix C.
5.1.1 Phase structure for small charges
In the region of small charges , as one can see from Fig. 13, for arbitrary temperature there are two branches of solutions for the black hole radius. These two branches correspond to small and large black holes, respectively. The small black holes have negative specific heat which is defined according to the formula
and therefore they are unstable. The large black holes have positive specific heat and consequently they are locally stable. Moreover, as can be seen in Fig. 13 (right panel) the free energy of the large black holes is smaller than the free energy of the small black holes and therefore the large black holes are thermodynamically favorable and dominate the thermodynamic ensemble.
5.1.2 Phase structure for intermediate charges
In the intermediate charge interval defined by , there are four branches, as shown in Figs. 14–17. These branches will be called branches of very small (VSBH), small (SBH), large (LBH), and very large (VLBH) black holes, respectively.
Below we describe the phase structure in the subintervals , and
. The differences between the three subintervals are that
in the first subinterval no phase transitions exist, in the second subinterval phase transitions of first and
zeroth order exist, and in the third subinterval only first order phase transition exists. Such subintervals with the corresponding
phase transitions are also present for pure Einstein-Born-Infeld-AdS black holes.
Intermediate charge subinterval
In this subinterval 111111By and we denote the two minima of the function as can be seen in Fig. 14.. The and dependences are shown in Figs. 14 and 15 and the phase structure is described as follows. For low temperature satisfying there are two solutions for the black hole horizon radius corresponding to LBH and VLBH. From the dependence shown in Fig. 14 one can conclude that the LBH are thermodynamically unstable since they have negative specific heat, while the VLBH have positive specific heat and therefore they are locally stable. At the origin of two new branches of solutions appears and for we have four black holes with the same temperature and different radii – VSBH, SBH, LBH, and VLBH. The VSBH and LBH are unstable since their specific heat is negative, while the SBH and VLBH have positive specific heat and consequently they are locally stable. At two of the branches coalesce and disappear, leaving only two branches for corresponding to unstable VSBH and locally stable VLBH.
Intermediate charge subinterval
In this subinterval . The and dependences are shown in Fig. 16 and the phase structure is described as follows. For we have two branches – unstable LBH and locally stable VLBH. At two new branches appear and for there are four branches corresponding to unstable VSBH, locally stable SBH, unstable LBH and locally stable VLBH. At two of the branches coalesce and disappear and we are left with only two branches of unstable VSBH and locally stable VLBH. From to the VLBH have the smallest free energy and they dominate the thermodynamical ensemble. At the situation changes and for temperatures in the interval from to the ensemble is dominated by the SBH which have the smallest free energy. At this point phase transition between VLBH and SBH occurs and it is zeroth order phase transition since the free energy is discontinuous at that point – the free energy jumps to a lower value121212We have studied the zeroth-order phase transition observed at that point with the off-shell formalism in Appendix C. . Such unusual phase transitions of zeroth order have been observed for example in [67, 68, 69].
As we said for temperatures in the interval the ensemble is dominated by the SBH. As we can see in Fig. 16 for arbitrary the ensemble is again dominated by the VLBH which have the smallest free energy. This means that at the point we observe a phase transition between the SBH and VLBH and it is classified as a first order phase transition since the free energy is continuous at and the first derivative of the free energy is discontinuous at that point.
Intermediate charge subinterval
In this subinterval . The and dependences are shown in Fig. 17 and the phase structure is described as follows. For there are two branches corresponding to unstable VSBH and locally stable SBH. At two new branches appear and for temperatures we have four branches corresponding to unstable VSBH, locally stable SBH, unstable LBH and locally stable VLBH. At the branches of SBH and LBH disappear and we are left with unstable VSBH and locally stable VLBH for . From to temperature the free energy of the SBH is the smallest and the SBH dominate the thermodynamic ensemble. At temperature the free energy of the VLBH becomes smaller than the free energy of the SBH and for all temperatures the VLBH dominate the thermodynamic ensemble. At the point we have a first-order phase transition from SBH to VLBH.
5.1.3 Phase structure for large charges
In the case of large charges defined by , there are two branches – small and large black holes, as Fig. 18 shows. The small black holes have negative specific heat and consequently they are unstable. The large black holes are locally stable since their specific heat is positive. For arbitrary temperature the free energy of the large black holes is smaller than the free energy of the small black holes. Therefore the large black holes dominate the thermodynamic ensemble.
5.2 Phase structure for small values of the parameter
The phase structure for small values of the parameter is much simpler because the function has no inflection point. Therefore the phase structure is qualitatively the same for all values of the magnetic charge . The functions and are plotted in Fig. 19 for and . Only two branches exist corresponding to unstable small black holes and locally stable large black hole. The large black holes have lower free energy and therefore they dominate the thermodynamic ensemble.
The functions and for are shown in Fig. 20 in the case of pure Einstein theory which corresponds to . This case is qualitatively the same as the case when the scalar field is nontrivial except for large charges which fulfill the inequality . For these values of and , when , the branch of small black holes disappear because of the occurrence of an extremal black hole (see for example the curve corresponding to in Fig. 20).
In the current paper we have constructed numerically new scalar-tensor black hole solutions in AdS space-time coupled to Born-Infeld nonlinear electrodynamics. The properties of the solutions have been thoroughly studied through a combination of analytical and numerical techniques. The black hole solutions in the considered class of scalar-tensor theories are determined uniquely by the asymptotic charges of the black holes – in our case the mass and the magnetic charge. An important property is that in comparison to the corresponding solutions in General Relativity, namely, the Einstein-Born-Infeld-AdS black holes, the black holes considered here have a simpler causal structure – they have neither inner nor degenerate horizons.
The thermodynamics and the phase structure in the canonical ensemble of the obtained black holes have also been studied. Their phase structure is similar to that of the Einstein-Born-Infeld-AdS black holes in GR and the difference is manifested at large charges. This means that in the case under consideration there are no extremal black hole solutions which actually leads to one additional branch of solutions in the thermodynamical phase-space structure for large values of the charge.
The thermodynamical phase-space structure depends on the Born-Infeld parameter and the magnetic charge . A critical value has been found numerically. For the phase diagram is the same for all values of the magnetic charge . There are two branches corresponding to unstable, small black holes and locally stable, large black holes. In this case no phase transitions are possible.
In the case of , the structure of the phase diagram changes with the variation of the magnetic charge and three charge intervals are observed – small, intermediate, and large charges. For small and large values of the magnetic charge no phase transitions are possible and the phase diagram has only two branches which corresponds to two phases: unstable, small black holes and locally stable, large black holes. The intermediate charges are the most interesting because phase transitions can exist in this case. The phases are four: VSBH, SBH, LBH, and VLBH. Among them two are locally stable – the SBH and the VLBH – and phase transitions between the two black holes are possible in some cases. In the intermediate charge subinterval first order phase transition between SBH and VLBH exists – for small values of the temperature () the SBH are thermodynamically favorable because they have lower free energy. When we increase the temperature we reach a certain value of the temperature where phase transition to VLBH occurs and for all temperatures above that value the VLBH are thermodynamically favorable. This phase transition is of first order because the first derivative of the free energy is discontinuous at that point. An important observation is that for a small intermediate charge subinterval two phase transitions exist. One of them is the first order phase transition we just described and the other one is the zeroth order phase transition. In this case for small values of the temperature () the only locally stable branch is the VLBH. At temperature another locally stable branch appears – SBH, which have the smallest free energy and dominate the thermodynamical ensemble. This means that for phase transition between VLBH and SBH exists and it is of zeroth order since the free energy is discontinuous at that point. When we increase the temperature the point is reached where the first order phase transition between SBH and VLBH, that was described above, occurs. For again the VLBH dominate the thermodynamical ensemble. For all the other intermediate charges phase transitions are not possible since the free energy of the VLBH is always the smallest. Similar zeroth and first order phase transitions for certain charge intervals are present also in the corresponding solutions in General Relativity.
Let us finish with prospects for future work. As in the asymptotically flat case , in AdS spacetimes the scalar-tensor theories with () coupled to nonlinear electrodynamics admit non-unique black hole solutions . From AdS/CFT perspectives these non-unique black hole solutions could be considered as second order phase transitions in the dual field theory living on the boundary. Detailed considerations will be given in a forthcoming paper . Phase transitions due to non-unique AdS black hole solutions in Einstein-Maxwell-dilaton gravity with appropriate dilaton coupling function have been obtained quite recently in . Let us note however that the situation in the case of scalar-tensor AdS black holes coupled to the nonlinear electrodynamics is rather different in comparison with the Einstein-Maxwell-dilaton gravity.
S.Y. would like to thank D. Uzunov for the discussions on phase transitions. D.D. would like to thank the DAAD for a scholarship and the Institute for Astronomy and Astrophysics Tübingen for its kind hospitality. D.D. would also like to thank J. Kunz, B. Kleihaus, and D. Georgieva for the helpful discussions.
This work was partially supported by the Bulgarian National Science Fund under Grants DO 02-257, VUF-201/06, and by Sofia University Research Fund under Grant No 074/2009.
Appendix A Black hole thermodynamics in scalar-tensor
In this appendix we would like to make some comments on the thermodynamics of the black holes in scalar-tensor theories. The definitions of entropy, temperature, and mass (energy) of the scalar-tensor black hole are naturally related to the Einstein frame. The reasons are the following. The Einstein-frame area of the horizon can be interpreted as entropy of the horizon since it is non-decreasing in classical physical processes. The entropy of the event horizon in the Einstein frame , for example, is proportional to one fourth of the area of the horizon. In the Jordan frame, however, the area of the event horizon may decrease in classical processes  and a generalized definition of entropy needs to be introduced. The proper definition is [72, 73, 74, 75]
This definition is also consistent with the Euclidean and the Noether charge method .
Using relation (2) we find that
the definitions of the entropy in both conformal frames are in agreement. In the last two equations quantities and are the determinants of the induced metrics on the horizon in the Jordan and in the Einstein frames, respectively.
Further, it can be proved that the standard definitions of black-hole temperature are invariant with respect to conformal transformations that are regular on the event horizon . Since the conformal transformations (2) in the present paper are regular on the horizon (and everywhere outside the horizon), the temperature in the Einstein frame coincides with the temperature in the Jordan frame.
The measure of the internal energy of compact objects in STT needs also to be specified. In the general case the masses of the black holes in the Einstein frame and in the Jordan frame do not coincide. The mass in the Einstein frame is positive definite, can only decrease by emission of gravitational (and scalar waves), and possesses other natural energy-like properties unlike the mass in the Jordan frame. In other words the mass in the Einstein frame is the true energy of the compact objects within the framework of the STT. For more details on the mass of compact objects in the STT we refer the reader to [77, 78, 79, 80, 38, 39]. For the particular solutions studied in the current paper, however, the scalar field decreases fast enough and the black-hole masses in both conformal frames are the same.
Appendix B Derivation of the free energy
The action with surface terms and counterterms taken into account is given by
where is the metric induced on the boundary and is the trace of the extrinsic curvature of while is the counterterm action. We have also defined and the boundary stress-energy tensor is given by
If the boundary geometry has an isometry generated by the Killing vector , is divergence free which gives the conserved quantity
associated with the closed surface in the boundary . In the case when , the conserved quantity is the mass.
where is the Ricci scalar curvature of the boundary metric . With this counterterm for the asymptotic metric
we have up to the leading order