Charged accelerating black hole in $f(R)$ gravity

We obtain a charged accelerating AdS black hole solution in $f(R)$ gravity and investigate its thermodynamic behaviour. We consider low-acceleration black holes that do not have an acceleration horizon and obtain the first law of thermodynamics for them. We further study the parameter space of charged slowly accelerating $f(R)$ AdS black holes before investigating the behaviour of the free energy in both the canonical and grand canonical ensembles. We find a generalization of the reverse isoperimetric inequality, applicable to black holes in $f(R)$ gravity, that indicates these black holes can become super-entropic relative to their counterparts in Einstein gravity.


I. INTRODUCTION
The well known vacuum C-metric was introduced in [1,2], and then investigated in [3,4]. Later on, a charged C-metric was discovered [5]. Mathematically, the Cmetric is a solution of the Einstein(-Maxwell) field equations; physically, it can be interpreted as two uniformly accelerating black holes dragged by forces originating from conical singularities [6]. An accelerating black hole with not only charge but also rotation was raised in [7].
The C-metric with a negative (positive) cosmological constant has been used as a test background to study gravitationally and electromagnetically radiative fields near timelike (spacelike) infinity [8,9]. However, in contrast to the flat or de Sitter (dS) C-metrics, the anti-de Sitter (AdS) C-metric can describe a pair of accelerated black holes only when the acceleration parameter A and the AdS radius l satisfy a certain restriction, originally thought to be A > 1/l [10]. It has recently been shown that this condition must be corrected to be a more stringent one [11]. If this condition is not satisfied (which will be the case for sufficiently small acceleration A), then the AdS C-metric describes an single uniformly accelerating black hole having an outer (and possibly inner) horizon [10][11][12].
For quite some time the thermodynamics of accelerating black holes have been poorly understood (and indeed avoided) in the literature, most likely because they generically have two horizons, except for the slow-acceleration case noted above. However it has recently been shown that it is possible to make sense of their thermodynamics in the slow-acceleration case, and a first law for these objects was derived [13]. Furthermore, by allowing the tension related to the conical deficits to vary, thermodynamics of the accelerating black hole was generalized to include a new potential named thermodynamic length [14]. A holographic interpretation of the thermodynam- * mingzhang@jxnu.edu.cn † rbmann@uwaterloo.ca ics for a slowly accelerating AdS black holes was subsequently given [11], and thermodynamic variables were found to be consistently calculated using both conformal and holographic methods. Conserved charges were obtained by choosing a canonical time coordinate and using a covariant phase space method in [15], as well as checked by the Christodoulou-Ruffini mass formula [16]. An explication of the thermodynamic parameters of accelerating AdS black holes with charge and rotation was studied in [17], and these results were used to obtain a remarkable snapping-swallowtail phase transition for the charged case at a certain pressure [18]. For the rotating case this particular pressure 'splits' into three values that have the effect of admitting reentrant phase transitions obtainable in two ways, either by varying the temperature at fixed pressure or varying the pressure at fixed temperature [19]. Until now, all extant C-metrics describing accelerating black holes are solutions of the equations of motion in Einstein's classical gravitational theory. It is of interest to find accelerating black hole solutions in modified gravity theories, such as the renowned f (R) gravity [20]. A number of reasons whose origins are in high-energy physics [21][22][23] and cosmology [24,25] motivate consideration of higher-order curvature invariants to the gravitational action. Amongst the various possibilities, f (R) gravity is an excellent candidate for two reasons. First, it encapsulates the features of the higher-order gravity by, for example, taking a simple form where R is the Ricci scalar and the α i are coefficients; other higher-order terms such as R µν R µν are not considered [20]. Second, it may be the only higher-derivative theory that avoids the Ostrogradski instability [26].
In this paper, we obtain charged accelerating black hole solutions in f (R) gravity and study their thermodynamics. The thermodynamics of charged black holes in f (R) gravity have been studied previously [27], and phase transitions analogous to the van der Waals liquid-arXiv:1908.05118v2 [hep-th] 22 Aug 2019 gas system between small and large black holes were observed to take place. In our extension to this case we find a number of new results. One is a generalization of the reverse isoperimetric inequality [28] that includes both conical deficits and the effects of f (R) gravity. We find that f (R) gravity can yield both sub-entropic and super-entropic black holes, depending on the magnitude of η = 1+f (R 0 ), where R 0 is value of the Ricci scalar for the accelerating solution. Furthermore, we find that the parameter space of allowed solutions is either enlarged or diminished, depending on the size of η. Our investigation of the phase behaviour of these black holes indicates that both the expected van der Waals transitions and snapping swallowtail behaviour observed previously [18] occur here as well, but we also find new behaviour in the fixedpotential ensemble, including Hawking-Page-type curves for unstable and stable black holes. We find that not only the lowest temperature attainable for the black hole but also the highest temperature that the small black hole can reach increases for increasing η. Furthermore, both kinds of temperatures have depend on the acceleration of the black hole, as we shall demonstrate.
We begin by introducing a charged accelerating black hole solution in f (R) gravity and calculating its thermodynamic quantities in Sec. II. We will analyze the parameter space and phase transition for the charged slowly accelerating f (R) AdS black hole in Sec. III. The final section is devoted to our summary.

A. Solution
The action of f (R) gravity containing Maxwell term is with Here R is the Ricci scalar, f (R) is an auxiliary function of R, F ab = ∇ a B b −∇ b B a , and B a is the gauge potential. The corresponding equations of motion are and with It is easy to check that the electromagnetic field is traceless, as Supposing that we want to get a maximally symmetry solution, we can know that the Ricci scalar must be constant [20]. When the curvature scalar is constant, i.e., R = R 0 (R 0 = 0), according to (4), we can obtain [29,30] Then (4) can be re-expressed as where we have defined According to the equation of motion (10), we can obtain a charged accelerating black hole solution where Ω = 1 + Ar cos θ, In the black hole solution, the conformal factor Ω determines the conformal boundary r b of the black hole and r b = −1/(A cos θ). A, m, q are individually the acceleration, the mass parameter and the electric parameter of the black hole. K stands for the conical deficits of the black hole on the north and south poles. The parameter α is used to rescale the time coordinate so that we can get a normalized Killing vector at the conformal infinity [11,12,17]. When A = 0 we recover the charged AdS black hole in f (R) gravity [31]. R 0 < 0, R 0 = 0 and R 0 > 0 correspond to asymptotically AdS, flat, and dS accelerating black holes, respectively. In the following, we will only investigate the case R 0 < 0. When A = 0, K = 1 R 0 < 0, the solution reduces to the f (R) black hole [29] for which we need η > 0 to assure the existence of inner and outer horizons [30]. For consistency, we will only consider the case η > 0 for the accelerating f (R) AdS black hole here. Then we can solve the equation of motion (5) for the gauge field to obtain the electromagnetic tensor where and r + is the outer horizon. When the solution is identical to the charged accelerating AdS black hole in Einstein gravity. Hereafter, we will write R 0 = −12/l 2 , so that the blackening factor of the black hole can be written as and we note that η and l are independent parameters.
B. Thermodynamic first law of slowly accelerating f (R) AdS black hole We will investigate slowly accelerating f (R) AdS black hole here. The condition of the slow acceleration will be discussed later. The mass of the slowly accelerating AdS black hole can be obtained by conformal prescription [32,33]. We can choose a conformal factorΩ = η −1 lΩr −1 to eliminate the divergence near the conformally infinite boundary. After a conformal transformation on the spacetime metric (12), we can obtain the conserved charge related to the Killing vector ξ a as whereN a is a normal covector of the conformal boundary determined byΩ,C a bcd is the Weyl tensor of the conformal metricḡ ab . After lettingΩ → 0 in the conformal metricḡ ab , we can obtain the spacelike area element as Then the mass M of the slowly accelerating f (R) AdS black hole can be calculated as It should be noted that we have chosen the conformal factor so that the resulting mass of the slowly accelerating f (R) AdS black hole becomes that of the f (R) black hole [30] calculated by the quasilocal approach [34][35][36].
In contrast to the slowly accelerating AdS black hole in Einstein gravity, the entropy of the accelerating AdS black hole in f (R) gravity cannot be simply calculated from the area law [37]; rather the Wald method [38] must be used. This gives whereĥ is the determinant of the induced metric on the t = const. and r = r + hypersurface, and ab is a normal bivector which satisfies ab ab = −2. The temperature of the black hole can be obtained as Then we can know The electric charge of the black hole is from Gauss' law. The conjugate electric potential is where n a is a normal covector on the conformal boundary determined by Ω, h is the determinant of the induced metric on the conformal boundary, β is related to the surface gravity κ at the event horizon r + by β = 2π/κ [39].
Note that although a rescaling of q renders the black hole solution (21) the same as that for the accelerated charged black hole in Einstein gravity [17], this parameter actually does not scale out of the thermodynamics. The reason is that that mass (25) and entropy (26) retain their η-dependence, and in fact the absence of η in (29) and (30) is essential in satisfying the first law. We have checked that the gauge potential (19) can not contain a factor 1/ √ η, or else (10) can not be satisfied. Consequently we disagree with the conserved electric charge and conjugate potential previously obtained for f (R) black holes [27,30], where there are respective factors of 1/ √ η and √ η.
We can also define [50] which respectively are average and differential conical deficits for the charged f (R) accelerating black hole. We note that as η → 0 we obtain a large average deficit ∆ and a vanishingly small differential deficit C; conversely for large η both ∆ and C approach their values in the charge neutral case. Since both µ ± are dimensionless, these quantities, along with their thermodynamic conjugates λ ± [14], do not enter the Smarr relation (34).
There is a direct but rather cumbersome method to obtain these quantities from the first law [17]. Rather than proceeding in this manner, we can instead write the mass entirely in terms of the extrinsic thermodynamic parameters, obtaining From this we can compute the temperature, electric potential and thermodynamic volume as which, upon using the expressions (26), (29), and (32) can be shown to agree with those in (27), (30) and (33) respectively.
The thermodynamic lengths conjugate to the tensions µ ± are therefore straightforwardly obtained and it is then easy to check that the first law is satisfied.
The key advantage of the formulation (41) is that it elucidates the chemical structure of the accelerating black hole [50]. Indeed, using (41) and (44) we obtain a generalization of the Reverse Isoperimetric Inequality [28,[51][52][53] 3V 4π for accelerating f (R) black holes, where A is the horizon area. The equality is saturated if C = 0, and reduces to that in Einstein gravity [50] if η = 1. A larger deficit yields a smaller ∆, and therefore a smaller entropy with respect to the volume, tending to make the black holes sub-entropic. However a larger value of η works in a contrary manner, allowing for a larger entropy (but smaller area!) relative to the volume, tending to make the black holes super-entropic [54,55]. We can confirm our result by calculating the action with Gibbons-Hawking boundary term and boundary counterterms [32,[56][57][58] I =I bulk + I surf + I ct where (49) In the bulk term, one should note that R + f (R) = −6η/l 2 , K is the extrinsic curvature scalar of the boundary surface, R is the intrinsic curvature of the conformal boundary described by the induced metric h ij . One should note that the boundary here is the conformal boundary. After calculation, one can obtain the Gibbs free energy G as We have neglected the boundary term corresponding to the electricmagnetic field in the action (48), which means that the system's electric potential is fixed but the electric charge is variable [59][60][61][62].

III. PHASE TRANSITION OF THE SLOWLY ACCELERATING f (R) AdS BLACK HOLE
A. Parameter space We will discuss permitted range of the parameters m, A, q, l, η for the slowly accelerating f (R) AdS black hole. Following convention in [18], we definẽ m = mA,q = qA,Ã = Al,r = r l , (52) which are dimensionless quantites, then N (r) and H(θ) can be written as First, we must ensure the existence of an outer event horizonr + such that withr + = r + /l, and the positivity of the blackening factor in the region between the outer horizon and the conformal boundary, which can be expressed as The condition (56) can be assured if we have Second, we should make sure which is equivalent to let where Θ is the unit step function and Θ(0) = 1/2. Third, we must eliminate the acceleration horizon. The extreme condition for the emergence of the acceleration horizon is from which we can getm =m(q, θ, η) andÃ =Ã(q, θ, η). Moreover, one should note that in (16), which is equivalent tõ The parameter space has been shown in Fig. 1 with white enclosed regions. We have chosen two different values for characteristic quantity η of the charged accelerating black hole in f (R) gravity. It is evident that a larger η is endowed with a larger physically reasonable parameter space. There is a point where the blue, green and red curves intersect (we name it point X, as the convention in [18]), which can be solved for, yielding Combining the expressions (57), (59), (60), (62) and (63) with the figures, when η → ∞, we obtain and whereas for η → 0 (note that η > 0 for charged accelerating f (R) black hole), we have and 0 <Ã <Ã X → 0 + .
(66) demands that |q| < 1. When all curves intersect into one point (we can name it point E), which makes the permitted region to be none, we can obtain the coordinate of the extreme point as When η = 1, the situation reduces to that studied previously [18] for charged accelerating black holes in Einstein gravity.

B. Phase transitions
We now consider the phase behaviour of these slowly accelerating black holes. We begin with the canonical ensemble, where the electric charge of the black hole is fixed. The Helmholtz free energy for this ensemble, which can be obtained by including (51) in the computation of the action, is Without loss of generality [18], we choose K = Ξ+2m, so that µ + = 0. The conical deficit is located entirely along the south pole axis θ = π, with µ − =m/K. According to (29), (32), (39), (40) and (63), the pressure corresponding to the point X is  from which we can see that the parameter η does not have a qualitatively different effect on the pressure of the intersection point X. As a result, when considering phase structure of the slowly accelerating f (R) AdS black hole, we see that it will be qualitatively similar to that in Einstein-AdS gravity [18].
In other words, a slowly accelerating f (R) AdS black hole will experience a snapping swallowtail process when the pressure of the black hole decreases below P X , and the black hole will experience zeroth order, first order, second order and reentrant phase transitions [18]. We depict the Helmholtz free energy in terms of the temperature in Fig. 2, from which we can see that the phase transition of the slowly accelerating f (R) AdS black hole is van-der Waals like for P > P X , and the branch corresponding to the small black hole with low temperature has snapped for P < P X .
Turning to the grand canonical ensemble, where the electric potential of the black hole is fixed, we plot the Gibbs free energy (50) in terms of the temperature to see the phase structure. Results are shown in Fig. 3 for fixed η (top diagram), fixed µ − (middle diagram) and fixed √ ∆η (bottom diagram). Setting C = 0 but ∆ = 1 (i.e., µ − = µ + = 0) we obtain the zero-acceleration f (R) black hole threaded by a cosmic string. In this case, by analyzing the derivative of the temperature with respect to the entropy we find that √ ∆η > Φ yields curves similar to those in the top diagram in Fig. 3, whereas √ ∆η Φ yields the Gibbs free energy monotonically decreasing with temperature (not illustrated here), a phenomenon noted previously for the zero-acceleration case without a cosmic string [63,64] . For ∆ = 1, C = 0 (i.e., µ − = µ + = 0) and √ η > Φ we obtain the uppermost (red) curve shown in the top diagram in Fig. 3, for which a standard Hawking-Page transition occurs at G = 0. We do not have an interpretation of what the transition at G = 0 means for µ − > 0. For the charged slowly accelerating f (R) black hole, we find that T (S) = 0 always has a solution for S > 0. As a result Hawking-Page(-like) curves for the Gibbs free energy are always present. From Fig. 3, we can see that the lowest temperature attainable by a charged slowly accelerating f (R) black hole increases for increasing µ − and η. We also find that there exist curves that at low temperatures have G > 0 on portions of both branches, whereas at high temperatures G < 0 on both branches. The lower branch corresponds to stable black holes when G < 0. The upper branches terminate at some finite large temperature that is not visible in the diagrams (except for the two terminal points shown in the bottom diagram of Fig. 3), and which increase for decreasing µ − and for increasing η.

IV. SUMMARY
We have obtained the first accelerating black hole in modified gravity (f (R) gravity), and have investigated its thermodynamics, computing its mass, entropy, temperature, electric charge, electric potential, and tensions. We find a number of similarities to the charged accelerating case studied previously in Einstein gravity [18].
The structure of its thermodynamic phase transitions are similar to those in Einstein gravity, and the 'snapping swallowtail' phenomenon is exhibited.
However we also find a number of distinctions due to the parameter η that arise because the mass and entropy are both linearly dependent on this quantity. This means that it cannot be scaled out of the charge parameter, despite superficial appearances to the contrary. We also obtain a generalization of the reverse isoperimetric inequality (47), suitable for f (R) gravity. While the dependence on the conical deficit is (as expected) the same as in Einstein gravity [50], we find that increasing η tends to allow the black holes to have more area relative to their thermodynamic volume, tending to make them super-entropic. Furthermore, the allowed region of parameter space governing the phase transitions becomes η-dependent, broadening the range of possibilities. We found that for increasing η the accessible parameter space is enlarged. Specifically, we gave the upper and lower bounds for the parametersm andÃ that characterize the parameter space when η varies between 0 and ∞. We have analyzed the phase structures of the charged slowly accelerating f (R) AdS black hole both in the canonical ensemble and grand canonical ensemble. In the fixedcharge ensemble, we found van der Waals phase transitions and snapping swallowtail behaviour, similar to that in Einstein gravity [18]. In the ensemble where the electric potential is fixed, we found that the lowest temperature the black hole can attain increases for increasing µ − and η. We also found that there are always two branches in the G − T diagram for √ ∆η > Φ, √ ∆η = Φ and even √ ∆η < Φ, where the upper one terminates at a high temperature. The terminated temperature decreases for increasing µ − and decreasing η.
f (R) gravity enables us to study the inflation and accelerated expansion epochs of our universe. It can uniformly explain the early-time inflation as well as the dark energy epoch [65]. It has been testified by many applications in cosmology and astronomy [66], it has also been constrained by realistic FRW cosmology [67] and stability consideration [68]. f (R) gravity is such a well-behaved modified theory that it can show the evolution of the universe and it can also give correct cosmological predictions [69,70]. It has been shown that f (R) theory without cosmological constant can reproduce the behaviour of Einstein-Hilbert action with cosmological constant [71].
The charged accelerating black hole in f (R) gravity should provide further insight into f (R) gravity. The role of η needs to be better understood. For example, for η > 1, will these black holes exhibit thermodynamic instabilities similar to those found recently [72] for other super-entropic black holes? It is easy to extend our solution for the charged accelerating black hole in f (R) gravity to one with rotation, and it is possible that interesting η-dependent phenomena may be present in this case. Likewise, the possibility that η itself could be a thermodynamic parameter merits investigation. Work on these topics is in progress.