Heat engines for dilatonic Born–Infeld black holes

In the context of dilaton coupled Einstein gravity with a negative cosmological constant and a Born–Infeld field, we study heat engines where a charged black hole is the working substance. Using the existence of a notion of thermodynamic mass and volume (which depend on the dilaton coupling), the mechanical work takes place via the pdV terms present in the first law of extended gravitational thermodynamics. The efficiency is analyzed as a function of dilaton and Born–Infeld couplings, and the results are compared with analogous computations in the related conformal solutions in the Brans–Dicke–Born–Infeld theory and black holes in anti-de Sitter space-time.


Introduction
Recent interest in treating the cosmological constant as a dynamical parameter [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15] has led to important extensions of the classical thermodynamic properties of a black hole [16][17][18][19], which relates the mass M, surface gravity κ, and outer horizon area A of a black hole solution to the energy, temperature, and entropy (U , T , and S, resp.) according to (in geometrical units where G, c,h, k B are set to unity) Now, the cosmological constant treated as pressure p = − /8π , has a conjugate variable, the thermodynamic volume V associated with the black hole. In this extended thermodynamics, temperature and entropy continue to be related to surface gravity and area as usual, while the mass, however, turns out to be related to enthalpy H [5]: M = H ≡ U + pV . The first law now reads dM = T dS + V d p. The black holes may have other parameters such as gauge charges, angular momentum, coupling constants (Gauss-Bonnet, Born-Infeld) which enter additively with their conjugates in the first law (1.2) in the usual way. For static black holes, the thermodynamic volume V is just the geometric volume (defined in terms of the horizon radius r + ) of the black hole in question [20], but, in general, the two volumes differ, leading to novel physics such as in rotating black holes, AdS-Taub-nut geometry, and black holes with dilaton fields (see for instance [7,[21][22][23][24][25][26]). An extended thermodynamical phase space treatment leads to an exact identification of small to large black hole phase transition in charged AdS and related black holes to a van der Waals liquid-gas phase transition [27,28], including an exact map of critical exponents. Furthermore, the phase transitions occur in the p-T plane as opposed to the Q-T plane and hence identical parameters are now being compared on both sides [29]. The possibility of extracting mechanical work from heat energy via the pdV term present in Eq. (1.2) has led to the proposal of a holographic heat engine in [30], where the working substance is a black hole solution of the gravity system. Several holographic engines have since been studied [31][32][33][34]. The black hole in particular provides an equation of state. Work can be extracted from such an engine by defining a cycle in state space where there is a net input heat flow Q H , a net output heat flow Q C , and a net output work W, such that Q H = W + Q C .
The efficiency of such heat engines can be written in the usual way as η = W/Q H = 1 − Q C /Q H . Its value depends crucially on the equation of state provided by the black hole and the choice of the cycle in state space. Considering the thermodynamical cycle in Fig. 1 advocated in [30,35,36] for static black holes, the entropy and the volume turn out to be dependent on each through r + . This means that isochores are adiabats, and hence the heat flows in the cycle in Fig. 1 occur only along top and bottom lines. Formal computation of the efficiency proceeds via the evaluation of C p dT along those isobars, where C p is the specific heat at constant pressure. This in general being difficult, efficiency was evaluated in various limits in [35,36]. An exact formula for efficiency, was later obtained in [37], using the fact that the mass of the black hole is just the enthalpy and total heat flow along an isobar is the change in enthalpy. For static black holes, M can be written as a function of p and V . Let us note that defining heat engines via cycles in state space (with dynamical cosmological constant) represents a journey through a family of holographically dual field theories [38][39][40][41][42] (at large N c ). The exact holographic dictionary corresponding to heat engines and their efficiency needs further study. Nevertheless, we restrict to applications to black holes in Einstein gravity with higher derivative corrections, which are interesting on their own right. Einstein gravity is considered to be an effective description of the underlying quantum gravity, such as string theory in the low energy limit. Thus, it is interesting to study the effect of stringy corrections on heat engines and their efficiency. The effect of Gauss-Bonnet and Born-Infeld higher curvature corrections on efficiency was analyzed in [35,36].
In this paper, we study the efficiency of heat engines in the presence of another stringy effect, namely, the dilaton field (Einstein gravity non-minimally coupled to a dilaton is present in the low energy limit of string theory [43]). In particular, we consider the Einstein-Maxwell dilaton system in the presence of two Liouville type potentials and also dilatonic black holes coupled to nonlinear Born-Infeld theory (Dilatonic-BI) in an extended phase space, in fixed charge ensemble [44]. We also consider its conformally consistent counter part, the Brans-Dicke Born-Infeld theory (BD-BI) studied in [45]. BD (Brans-Dicke) theory has been significant in the explanation of the cosmic inflation [46], and consistent with Dirac's large number hypothesis and Mach's principle [47,48]. Thermodynamics of charged black holes in Brans-Dicke theory have been studied in [49][50][51][52]. This theory produces the solar system experimental observations with a specific domain of BD parameter ω [53].
Moreover, the presence of the dilaton field in Einstein-Maxwell theory changes the causal structure of the spacetime and modifies the thermodynamic properties of the black holes in a non-trivial way. Rich structure and p-V criticality in black holes with higher derivative couplings, Born-Infeld and dilaton black holes have been studied earlier. 1 In the case, when there are Liouville type potentials for the scalar fields, the solution is asymptotically neither flat nor AdS. Furthermore, an exponential or Liouville potential represents the higher-dimensional cosmological constant present in non-critical string theories [74][75][76][77][78][79], non-trivial curved adS backgrounds [80], or leading g s corrections to critical string theories in a flat background. Starting from standard AdS/CFT duality in higher dimensions, holography of models with Liouville type scalar potentials are generated upon dimensional reduction procedure [81]. In [82], it has been conjectured that the linear dilaton space-times, which arise as nearhorizon limits of dilatonic black holes, might exhibit holography.
Non-asymptotically flat/AdS black hole space-times have been actively studied for possible extensions of holography [82][83][84][85][86][87][88][89][90][91][92][93][94][95][96]. The usual notions of thermodynamic mass, thought of as enthalpy of a space-time, akin to an AdS black hole go through for more general backgrounds. For instance, in the context of black holes in Liftshitz space-times (which are asymptotically neither AdS nor flat) [97,98], it has been argued that introducing pressure 2 together with thermodynamic volume and studying extended thermodynamic phase structure (in spite of the fact that all thermodynamic quantities now depend on the dilaton coupling constant) is physically and holographically sensible, with applications to condensed matter systems and quantum criticality. A holographic interpretation for the Van der Waals transition was proposed in [30], within the extended phase thermodynamics, where varying the cosmological constant in the bulk corresponds to perturbing the dual CFT, triggering a renormalization group flow. The transition is then interpreted not as a thermodynamical transition but, instead, as a transition in the space of field theories. Having scalar fields in the bulk (such as the charged Dilaton system in the present manuscript and other examples [33,97]) might turn on certain operators in the boundary theory triggering a non-trivial RG flow. In particular, there might be solutions of dilatonic theories with Liouville type poten-tials connecting the IR dynamics to AdS asymptotics in UV [99].
Also, pressure for asymptotically non-flat/ads black holes with Liouville type potentials has been introduced and the corresponding PV criticality studied in good detail in [68] and extended to conformally coupled scalars, i.e., the Brans-Dicke-Born-Infeld solutions [45]. Following these works, and the existence of an extended first law with pressure and volume, including the presence of PV criticality allows us to naturally define a heat engine, exactly as in the examples considered for AdS, leading to extension of the results of [30,44,45,68]. The working substance is still the charged black hole, however, the efficiency of heat engines will now depend on the coupling constants provided by dilaton and Born-Infeld theories. A feature of our heat engines is that the volume depends on the coupling constant of dilaton (α) and electromagnetic fields (β) and is not same as the geometrical volume. Existence of an exact formula allows us to study efficiency as a function of both couplings, i.e., η = η(α, β) and take various limits where we keep α fixed while tuning β and vice versa. In particular, in the limit α → 0 and in the high temperature limit, our exact results agree with the effect of Born-Infeld field on efficiency, captured in [36].

Heat engines from charged black holes in Dilatonic and Brans-Dicke-Born-Infeld theories
Following the discussion of last section, where a cycle in state space was presented for heat engines from charged black holes, we continue with the computation of the efficiency of such engines. We first study the dilatonic Born-Infeld model and later study the corrections to efficiency of heat engines in Brans-Dicke-Born-Infeld model.

Dilatonic Born-Infeld model
For the purpose of computing efficiency, we start from the relevant expression for mass of the Born-Infeld dilaton black hole [44] (details of black hole solutions are reproduced in "Appendix A" for reference), where b is an arbitrary positive constant, α is the dilaton coupling constant, n represents the number of spatial dimensions (we restrict to n > 3) and ω n−1 is the volume of the constant curvature hypersurface characterizing the horizon . Using the expression for m (see "Appendix A"), the mass can be expressed in terms of other thermodynamic parameters as A few comments are in order. Here, p is the pressure and β is the Born-Infeld parameter, where β → ∞ corresponds to the Maxwell limit. γ = α 2 /(α 2 + 1) and and r + is the radius of the horizon. k(> 0) is constant characterizing the (n − 1) dimensional hypersurface. The temperature expressed as a function of other thermodynamic parameters is The thermodynamic volume V is different from the geometrical volume due to the dependence on γ [44], Now the equation of state p(V, T ) for our working substance in the p-r + plane, or equivalently the p-V plane (using Eq. (2.4)) is [44], . A possible scheme for our heat engine (based on the cycle 3 given in Fig. 1) involves specifying values of temperatures (T 2 , T 4 ) (which in turn fixes (T H , T C ) ) and volumes (V 2 , V 4 ). The pressures p 1 = p 2 and p 4 = p 3 have to be computed from the equation of state and depend on the couplings α and β. Since the radii r 1 , r 3 can be obtained analogously and the mass M, written as a function of r + and p is as in Eq. (2.2), the efficiency of the engine can now be studied as a function of couplings α and β. Considering the cycle given in Fig. 1, the efficiency of heat engines can be defined entirely in terms of the black hole mass evaluated at the corners as given in (1.3). Notice that in the present scheme the Carnot efficiency η C = 1 − T C T H , the upper bound to our engine efficiency working between highest and lowest temperatures T H and T C , respectively, is fixed for all α, β. Another useful quantity to compare with is the efficiency in the Einstein-Maxwell case η 0 . 4 We first check the case where the dilaton coupling α is set to zero, in which case we have a pure Born-Infeld black hole. The efficiency of the heat engine in this case was studied in [36], in the high temperature limit. For n = 4, efficiency (Eq. (1.3)) takes the form For large p 1 , one obtains (2.9) 4 (i.e., the limit α → 0 and β → ∞, and we also rescaled the charge q → (n−1)(n−2) 2 q to get an exact Reissner-Nordstrom-anti-de Sitter black hole).
In fact, for n = 3, for large volume branch of solutions and neglecting q to leading order, we have This matches with equation (20) in [30]. From Fig. 2, it can be seen that both efficiency ratios, i.e., η/η C and η/η 0 , grow slowly for a while and then rise, in agreement with the results in [36] for high temperatures. We see from Fig. 3 that, an increase in q causes significant changes in the efficiency. In fact, we can see the effect of various parameters 5 on efficiency from Fig. 4.
We now keep α non-zero and study the resulting efficiency in the limit β → ∞. Now η (Eq. (1.3)) for n = 4, can be expressed as For large p 1 , one obtains . The efficiency of our engine now depends only on the dilatonic coupling and a comparison with both η C and η 0 is again possible. Using the equation of state one can check whether the pressures ( p 1 , p 3 ) in the engine remain physical as α changes. Since we have fixed (T 2 , V 2 ) and (T 4 , V 4 ), the pressures are now α-dependent. In fact, the pressures become negative as α increases beyond 2, diverging at α = 1, 2 since the black hole solution is diverging at these points. If we consider the critical behavior of our black hole [44], the universal ratio ρ c is positive, provided α < 1, so we restrict ourselves to the physical range of α, i.e., 0 < α < 1 (see Fig. 5).
A study of how the efficiency η varies with respect to α shows that, initially, it falls as compared to η C , but then rises again (see Fig. 6). One notes that as α → 1, results on the efficiency are less reliable as pressure may no more be positive. Figure 7 shows a similar effect of various parameters on efficiency of the Born-Infeld case 6 6 The plots for η ηc vs α for different q (= 2, 25, 70, 100) are similar to Fig. 6.
When we consider the effect of both couplings (α, β) on the efficiency of our engine (see Fig. 8), for a sample range of parameters 10 −2 < β < 10 2 and 0 < α < 1 (we checked that the pressures are physical over this sample range of parameters), both ratios η/η C and η/η 0 decrease rapidly in the turnaround region where, roughly, 10 −2 < β < 10 −1 and become steady as β increases, while as α increases, initially both the ratios decrease up to α ≈ 0.91 and then raise again.

Brans-Dicke-Born-Infeld model
Computation of corrections to efficiency in the Brans-Dicke Born-Infeld theory proceeds by writing down the relevant expression for enthalpy and equation of state [45] (details of black holes and thermodynamic quantities are summarized in "Appendix B") as (2.14) Here, r + is related to the thermodynamic volume V as , −η , Now, to study the efficiency of the engine we cast our rectangular cycle in the Einstein frame as well as in the Jordan frame. Hereafter, for simplicity, we take β → ∞.
The behavior of pressures in both frames can be seen from Fig. 9, which shows the rapid fall of pressures in Jordan frame. Moreover, the pressure of the isobar in the Einstein frame is higher than the pressure of the corresponding isobar in Jordan frame. In fact, the height of the cycle ( p 1p 3 ) is higher for the Einstein frame which leads to more work. From Fig. 10, we can see that as α increases, in Einstein frame, the inflow of heat monotonously decreases while the work done decreases up to α = 0.97 before raising. The efficiency decreases slowly up to α = 0.9 before a rapid rise. In the Jordan frame, Q H again decreases monotonously while W decreases up to α = 0.96 then raises. The efficiency shows a similar behavior to the Einstein frame, however the minimum efficiency occurs at α = 0.91. Regardless of frames, maximum values of Q H and W occur at α = 0, whereas efficiency reaches higher values when α → 1. Indeed, Q H Max = 87089 and W Max = 34499.7.
For comparison, we plotted Q H , W and η in Fig. 11. It can be seen that, for a given value of α, engine running in Einstein frame takes more heat and generates more work and is also more efficient as compared to engine run in Jordan frame. This is so because the enthalpies (expressed in r + , p) are not the same for each frame (although the expressions for the mass are the same). At a given value of α, the enthalpy in BD theory dominates over the Einstein theory at the same pressure (Fig. 12), as well as at the same horizon radius r + (Fig. 12). This implies that at a given (r + , p), the enthalpy is higher in BD theory than in the Einstein theory.
Although BD theory dominates in enthalpy over Einstein theory, if we evaluate the enthalpies at the corners of the cycle, Einstein theory dominates over BD theory. This is because for a given volume, horizon radius r + of the black hole is large in Einstein frame than that in the Jordan frame (see Fig. 13). Also, for a given (V, T ), the pressure is higher in the Einstein frame than that in the Jordan frame. Since the equations of state (expressed as p(V, T )) are not same for both frames (though Hawking temperatures have the same expressions).
In both frames, we find that the net inflow of heat Q H , work W and efficiency η increase with n (see Figs. 14, 15), while the allowed range of α decreases from the upper bound when regulated with Carnot efficiency η C . In fact the window of the allowed values of α is wider in the Einstein frame than that in Jordan frame (see Table 1).

Remarks
We studied the effect of dilaton and Born-Infeld couplings on the efficiency of the holographic heat engines in Einstein gravity (with negative cosmological constant), where charged black hole is the working substance, in spite of the dependence of thermodynamic volume on dilaton coupling [68] and unusual asymptotics [87], pdV terms exist [45] and mechanical work is extracted via the pdV terms present in the first law of extended gravitational thermodynamics with a dynamical cosmological constant. In the case where the dilaton coupling is absent, our exact result agrees with the high temperature calculation in [36]. As seen from Fig. 8, this behavior continues to hold even for non-zero values of dilaton coupling constant, signifying that the variations in the efficiency with β are several orders of magnitude lower than that with α. A similar feature was also noticed in the context of heat engines in Gauss-Bonnet black holes [36].
Increasing the parameter q effects the efficiency significantly as seen from Fig. 3. We noticed in both Born-Infeld and dilaton cases that the increase in charge q and volume V 2 lowers the efficiency, whereas the ratio η η C approaches unity on the account of increase in temperature T 2 . Also for large p 1 , leading behavior of the efficiency is (1 − p 4 p 1 ). In fact, increase in q, V 2 and T 2 implicitly changes the height of the cycle p ≡ p 1 − p 4 , which changes the efficiency accordingly [100].
We also compared the efficiency of engines in dilatonic Born-Infeld theory and Brans-Dicke-Born-Infeld theory. We see that our engine produces more work in the Einstein frame than in the Jordan frame. The Einstein frame provides a longer cycle along with high pressures for the corresponding isobars in the Jordan frame. Though black hole possesses more enthalpy in Brans-Dicke theory, for a fixed volume, the horizon radius is larger for an Einstein black hole. Hence, the calculation of Q H and efficiency η as a function of enthalpies evaluated at the corners of the cycle yields larger values in the Einstein frame. We find that irrespective of the frames, the maximum values for Q H and W occur at α = 0, while efficiency η reaches to higher values when α → 1. We also checked that the qualitative behavior of the efficiency (as well as Q H , W ) does not alter in higher dimensions, although, the allowed range of α is small. For both the dilatonic Born-Infeld and Brans-Dicke-Born-Infeld cases, we choose a scheme where highest and lowest temperatures are held fixed to have a get close to the scheme independent answer. In this case, a comparison of our efficiency with two standards, i.e., the case η 0 (Einstein-Maxwell theory) and the Carnot efficiency η C could be performed. However, it would be nice to study the behavior of the efficiency in other possible schemes, as the equation of state still depends on coupling constants of the model and similar behavior is a priori not guaranteed. It would also be nice to have a better holographic understanding of heat engines which have been studied thus far, both with and without dilaton couplings and/or including other higher order gauge/gravity corrections to Einstein Gravity (whether asymptotically AdS/flat or not). An engine operating at the critical point could show further interesting scaling properties [101], especially in the large charge limit [102]. We leave these issues for future work.

A Appendix
The (n + 1)-dimensional (n ≥ 3) action in which gravity is coupled to a dilaton and Born-Infeld fields is [23][24][25][26]44] where R is the Ricci scalar curvature, is the dilaton field and V ( ) is a potential for : where 0 , , ζ 0 and ζ are constants and the Born-Infeld L(F, ) part of the action is given by Here, α is a constant determining the strength of the coupling of the scalar and electromagnetic fields, is the electromagnetic field tensor, and A μ is the electromagnetic vector potential. β is the Born-Infeld parameter with the dimension of mass. A general solution is where h i j is a function of coordinates x i which spanned an (n − 1)-dimensional hypersurface with constant scalar curvature (n − 1)(n − 2)k. Here k is a constant characterizing the hypersurface (consider k > 0). The electromagnetic field (finite at r = 0) and charge are given, respectively, as where q is an integration constant related to the electric charge of the black hole and ω n−1 represents the volume of constant curvature hypersurface described by h i j dx i dx j . R(r ) and (r ) are given, respectively, as where b is an arbitrary constant, γ = α 2 /(α 2 + 1). The ADM (Arnowitt-Deser-Misner) mass m of the black hole expressed in terms of the horizon radius r h is Finally, f (r ) is given as The temperature and entropy of the black hole are given, respectively, as where η + = η(r = r + ). The gauge A t and electric potential U , measured at infinity with respect to the horizon are respectively.

B Appendix
Einstein-BI-dilaton gravity and its Brans-Dicke counterpart The action of (n + 1)-dimensional BD theory, in which the dilaton field is decoupled from the matter field (electrodynamics) and coupled with gravity can be written as [45] where L(F) is the Lagrangian of BI theory R is the Ricci scalar, ω is the coupling constant, denotes the BD scalar field and V ( ) is a self-interaction potential for . Indeed, the BD-BI theory is conformally associated with the Einstein-BI-dilaton gravity. The appropriate conformal transformation is as follows: By means of this conformal transformation, one finds that the action of BD-BI transforms to the well-known dilatonic-BI gravity as where the potential V and the BI-dilaton coupling Lagrangian L F, are, respectively, and