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 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. Efficiency is analyzed as a function of dilaton and Born-Infeld couplings, and 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 Λ as a dynamical parameter [1]- [15] have 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, , k B are set to unity): Now, the cosmological constant treated as pressure p = −Λ/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, mass, however, turns out to be related to enthalpy H [5]: M = H ≡ U + pV . The First Law now reads: dM = T dS + V dp. (1. 2) The black holes may have other parameters such as gauge charges, angular momentum, coupling constants (Gauss-Bonnet, Born-Infeld etc.,) which enter additively with their conjugates in the First Law (1.2) in the usual way. For static black holes, 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]). 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 Van der Waals liquid-gas phase transition [24], including an exact map of critical exponents. Furthermore, the phase transitions occur in the p − T plane as opposed to Q − T plane and hence identical parameters are now being compared on both sides [25]. The possibility of extracting mechanical work from heat energy via the pdV term present in eqn. (1.2) has led to the proposal of a holographic heat engine in [26], where, the working substance is a black hole solution of the gravity system. Several holographic engines have since been studied [27][28][29][30]. 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 for heat engines 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 cycle in state space. Considering the thermodynamical cycle in figure (1) advocated in [26,31,32] 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 figure (1) occur only along top and bottom lines. Formal computation of 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 [31,32]. An exact formula for efficiency was later obtained in [33], using the fact that mass of the black is just the enthalpy and total heat flow along an isobar is 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 [34][35][36][37][38] (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 [31,32].
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 dilaton is present in the low energy limit of string theory [39]). 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 [40]. We also consider its conformally consistent counter part, the Brans-Dicke Born-Infeld theory (BD-BI) studied in [41]. BD (Brans-Dicke) theory has been significant in the explanation of the cosmic inflation [42], and consistent with Dirac's large number hypothesis and Mach's principle [43]. Thermodynamics of charged black holes in Brans-Dicke theory have been studied in [44][45][46][47]. This theory produces the solar system experimental observations with a specific domain of BD parameter ω [48].
Moreover, the presence of the dilaton field in Einstein-Maxwell theory changes the causal structure of the space-time 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 [55,56], non-trivial curved adS backgrounds [57], 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 [58]. In [59], it has been conjectured that the linear dilaton spacetimes, which arise as near-horizon limits of dilatonic black holes, might exhibit holography.
Non-asymptotically flat/AdS black hole spacetimes have been actively studied for possible extensions of holography [59][60][61][62][63][64][65][66][67][68]. 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) [69,70], 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 [26], 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 [29,69]) 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 potentials connecting the IR dynamics to AdS asymptotics in UV [71].
Also, since pressure for asymptotically non-flat/ads black holes with Liouville type potentaials has been introduced and the corresponding PV criticality studied in good detail in [52] and extended to conformally coupled scalars, i.e., the Brans-Dicke-Born-Infeld solutions [41]. 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 [26,40,41,52]. 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 [32].

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 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 [40](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), mass can be expressed in terms of other thermodynamic parameters as 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. Temperature expressed as a function of other thermodynamic parameters is, The thermodynamic volume V is different from the geometrical volume due to dependence on 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 [40], . A possible scheme for our heat engine (based on the cycle 3 given in figure 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 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 equation (2.2), the efficiency of the engine can now be studied as a function of couplings α and β. Considering the cycle given in figure 1, 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 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. Efficiency of the heat engine in this case was studied in [32], in the high temperature limit. For n = 4, efficiency (equation (1.3)) takes the form as For large p 1 , one obtains, In fact, for n = 3, for large volume branch of solutions and neglecting q to leading order, we have This matches with the equation (20) in [26].
From figure (2), it can be seen that both the 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 [32] for high temperatures. We see from figure (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 figure (4). 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-deSitter black hole) 5 while varying the parameters one must check the validity of pressures.   We now keep α non zero and study the resulting efficiency in the limit β → ∞. Now η (equation (1.3)) for n = 4, can be expressed as For large p 1 , one obtains which shows the leading behavior of η is . 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, 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 [40], 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 figure 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 figure 6). One notes that as α → 1, results on the efficiency are less reliable as pressure may no more be positive.
When we consider the effect of both the couplings (α, β) on the efficiency of our engine (see figure 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 the 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. holes and thermodynamic quantities are summarized in Appendix B) as:

Brans-Dicke-Born-Infeld Model
(2.14) Here, r + is related to the thermodynamic volume V as , dilatonic BI, , BD-BI.  where, 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 figure (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 1 − p 3 ) is more for the Einstein frame which leads to more work.
From figure (10), we can see that as α increases, in Einstein frame, the inflow of heat monotonously decreases and the work done decreases up to α = 0.97 then raising, while the efficiency is decreasing slowly up to α = 0.9 before a rapid rise. Whereas, in Jordan frame, Q H is again monotonously decreasing and W is decreasing up to α = 0.96 then raising while the efficiency shows similar behavior with 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 to higher values when α → 1. Indeed, Q H M ax = 87089 and W M ax = 34499.7. For comparison, we plotted Q H , W and η in figure (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 same for each frame (although the expressions for mass are same). At a given value of α, the enthalpy in BD theory dominates over the Einstein theory at the same pressure (figure 12a), as well as at the same horizon radius r + (figure 12b). This implies that at a given (r + , p), enthalpy is more 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 figure 13). Also, for a given (V, T ), the pressure is more in the Einstein frame than that in the Jordan frame. Since, the equation of states (expressed as p(V, T )) are not same for both frames (even both have the same expression for Hawking temperature).
In both frames, we find that the net inflow of heat Q H , work W and efficiency η increase with n (see figures 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 Einstein frame than that in Jordan frame (see Table 1).   hole is the working substance, in spite of the dependence of thermodynamic volume on dilaton coupling [52] and unusual asymptotics [62], pdV terms exist [41] 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 [32]. As seen from figure (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 [32]. Increasing the parameter q effects the efficiency significantly as seen from figure (3). We noticed in both Born-Infeld and dilaton cases that the increase in charge q and volume V 2 lowers the efficiency, where as, 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 [72].
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 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 Einstein black hole. Hence, the calculation of Q H and efficiency η as a function of enthalpies evaluated at the corners of 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 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, 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 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 [73], especially in the large charge limit [74]. We leave these issues for future work.

A Appendix
The (n + 1)-dimensional (n ≥ 3) action in which gravity is coupled to dilaton and Born-Infeld fields is [23,40]: where R is the Ricci scalar curvature, Φ is the dilaton field and V (Φ) is a potential for Φ: where Λ 0 , Λ, ζ 0 and ζ are constants and the Born-Infeld L(F, Φ) part of the action is given by L(F, Φ) = 4β 2 e 4αΦ/(n−1) Here, α is a constant determining the strength of coupling of the scalar and electromagnetic fields, F 2 = F µν F µν , where F µν = ∂ µ A ν − ∂ ν A µ 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 ij 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 ij dx i dx j . R(r) and Φ(r) 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 where Υ = (n − 3)(1 − γ) + 1, and respectively.

B Appendix
Einstein-BI-dilaton gravity and its Brans-Dicke counterpart: The action of (n + 1)-dimensional BD theory, in which dilaton field is decoupled from the matter field (electrodynamics) and coupled with gravity can be written as [41] where L(F) is the Lagrangian of BI theory R is the Ricci scalar, ω is the coupling constant, Φ denotes the BD scalar field and V (Φ) is a self-interaction potential for Φ.
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 Φ and the BI-dilaton coupling Lagrangian L F , Φ are, respectively,