Gravity Effects on Charged Accelerating AdS Black Holes Using Holographic Tools

We investigate numerically f ðRÞ gravity effects on certain AdS/CFT tools including holographic entanglement entropy and twopoint correlation functions for a charged single accelerated Anti-de Sitter black hole in four dimensions. We find that both holographic entanglement entropy and two-point correlation functions decrease by increasing the acceleration parameter A, matching perfectly with literature. Taking into account the f ðRÞ gravity parameter η, the decreasing scheme of the holographic quantities persist. However, we observe a transition-like point where the behavior of the holographic tools changes. Two regions meeting at such a transit-like point are shown up. In such a nomination, the first one is associated with slow accelerating black holes while the second one corresponds to a fast accelerating solution. In the first region, the holographic entanglement entropy and two-point correlation functions decrease by increasing the η parameter. However, the behavioral situation is reversed in the second one. Moreover, a cross-comparison between the entropy and the holographic entanglement entropy is presented, providing another counterexample showing that such two quantities do not exhibit similar behaviors.

The entanglement entropy, which may be viewed as the von-Neumann entropy reflecting the entanglement between two different subsystems A and B, can be applied to the black hole. It has been considered also as a candidate not only for the statistical origin of the black hole entropy [8] but also for solving the information puzzle in black hole physics [9]. It is worth noting that the entanglement entropy acquires a similar geometric description to the bulk thermal entropy using holography [10][11][12]. Unfortunately, many attempts that have been made to explain the BH entropy in terms of quantum entanglement suffer from conceptual and technical difficulties. Indeed, the first difficulty is associated with the usual statistical interpretation of the black hole entropy in terms of microstates. This is conceptually very different from the entanglement entropy measuring the observer's lack of information by regarding the system as quantum states in an inaccessible region of the spacetime [13,14]. The second one comes from the fact that the entanglement entropy depends on the number of species of matter fields. The associated entanglement should reproduce the BH entropy. It depends also on the value of the ultraviolet (UV) cut-off, which may be needed to regularize the divergences corresponding to sharp boundaries between the accessible and inaccessible regions of spacetime. However, the Hawking-Bekenstein entropy should be universal. In the context of the extended phase space, it has been shown that the entanglement entropy undergoes qualitatively a similar behavior as entropy [15]. In paralleled ways, many investigations have been devoted to such a similitude [16][17][18][19]. However, it has been revealed that behavioral coincidence is associated with the first law of black hole thermodynamics. This can also be supported by the fact the entanglement entropy depends on the black hole mass [20,21]. It turns out that the holographic entanglement entropy is not dual to the black hole one. It should not be expected that they present the same behavior as observed in [20,21].
On the other hand, the inflation theory and dark energy can be exploited to build Einstein modified gravity. In this way, f ðRÞ gravity can be considered a good example to approach such modified theories. Adding f ðRÞ as higher powers of R, one can implement the Ricci and Riemann tensors and their derivatives in Lagrangians, which could describe possible physical models [22][23][24][25][26][27][28][29][30][31]. In this context, considerable efforts have been made to investigate the AdS black hole thermodynamics in the background of f ðRÞ gravity with constant curvature [32][33][34] as in the context of the AdS/CFT conjecture [34][35][36]. Recently, several results have been elaborated and refined to support such investigation lines, including accelerating black holes. The latters, being known to be described by the so-called C-metric [37][38][39][40], have been used not only to study the pair creation of black holes [41], the splitting of cosmic strings [42,43], but also to construct the black rings in five-dimensional gravity, motivated by string theory and related inspired models [44]. The thermodynamics of accelerating black holes [45], generalized results to the case of varying conical deficits for C-metric, and their applications to holographic heat engines, have been studied in many places, see for instance [45][46][47][48][49][50].
The aim of this paper is to contribute to these activities by investigating numerically the effect of the f ðRÞ gravity on certain AdS/CFT tools including holographic entanglement entropy and two-point correlation functions for a charged single accelerated Anti-de Sitter black hole in four dimensions. It has been shown that both holographic entanglement entropy and two-point correlation functions decrease by increasing the acceleration parameter A, matching perfectly with the literature [51,52]. Taking into account the f ðRÞ gravity parameter η, the decreasing scheme of the holographic quantities persist. However, it has been found a critical-like point where the behavior of the holographic tools changes. Two possible regions intersecting at such a transition-like point are observed. The first one corresponds to slow accelerating black holes while the second one is associated with a fast accelerating solution. In the first region, the holographic entanglement entropy and two-point correlation functions decrease by increasing the η parameter. However, the second one corresponds to a reversed behavioral situation.
The organization of the paper is as follows. We first review, in Section 2, the essential of the solution describing charged single accelerated AdS Black holes in f ðRÞ gravity background and the associated relevant thermodynamics. In Section 3 and Section 4, we, numerically, study the entanglement entropy and two-point correlation functions and we prob the effect of the f ðRÞ gravity parameter η on such holographic quantities. The last section is devoted to the conclusion and some open questions.

Charged Accelerating fðRÞ Black Hole Background
To start, one reconsiders the study of the physical content of f ðRÞ gravity. According to [53], its action containing a Maxwell gauge term takes the following general form In this action, L can be expressed as where R is the Ricci scalar. It is noted that f ðRÞ is an auxiliary function of R, which determines the gravity model [54,55]. The field strength F ab = ∇ a B b − ∇ b B a is derived from the gauge potential one form B a . Using field theory technics associated with the variation with respect to the metric, and the gauge field, one can get field equations of motion. Roughly, they are listed as follows and where one has used One can check that the electromagnetic field T ab is traceless It turns out that a maximally symmetry solution can be obtained by considering a constant Ricci scalar [56]. In the situation given by R = R 0 ðR 0 ≠ 0Þ, the metric field equation reduces to [32,57].
In this way, Eq. (3) can be rewritten as follows Advances in High Energy Physics Here, η is defined as Exploiting the results reported in [53] and using the equation of motion given by Eq. (9), we can derive a charged accelerating black hole solution based on the following line element The terms involving such a line element are given by In such a black hole solution, the conformal factor Ω denotes the conformal boundary r b of the black hole given by r b = −1/ðA cos θÞ. A, m, and q indicate individually the acceleration, the mass parameter, and the electric charge parameter of the black hole, respectively. K is the conical deficits of the black hole on the north and south poles. The present space-time involves conical singularities localized at (θ = 0, π). In this way, the metric regularity condition at the poles, θ + = 0 and θ − = π imposes that Generally, K is explored to regularize one pole producing a conical deficit or a conical excess along the other pole. Due to a negative energy corresponding to the source of a conical excess, it has been supposed, throughout this study, that the black hole is regular on the north pole where θ = 0 and K = K + . It turns out that a conical deficit can exist on the other pole for θ = π.
In order to get a normalized Killing vector at the conformal infinity [58][59][60], the parameter α can be exploited to rescale the time coordinate. When A goes to zero, it is observed that one can recover the usual charged AdS black hole in the f ðRÞ gravity [61].
It is noted that R 0 < 0, R 0 = 0 and R 0 > 0 are associated with asymptotically AdS, flat, and dS accelerating black holes, respectively. In the present investigation, we will be only interested in the case R 0 < 0. For A = 0, K = 1 and R 0 < 0, the solution reduces to the f ðRÞ black hole [32]. In this way, the condition η > 0 is needed to insure the existence of inner and outer horizons [57]. For certain physical reasons, we will only deal with the case η > 0 for the accelerating f ðRÞ AdS black holes. We hope other nontrivial cases could be dealt with in future works. Solving the equation of motion (4) for the gauge field B a , one can obtain the electromagnetic tensor The corresponding calculation gives where r + (the complete form of the r + is given in the appendix) is the outer horizon. It is worth noting that for the solution coincides with the charged accelerating AdS black hole in the Einstein gravity framework. For R 0 = −12/ℓ 2 , the blackening factor of the black hole can be written as where η and ℓ are considered as independent physical parameters.

Holographic Entanglement Entropy of Charged Accelerating Black Holes in fðRÞ Gravity Background
In the present investigation, we ignore the acceleration horizons of C-metrics and consider only the black hole horizon in order to get a well-defined temperature. This kind of simplification is known as the "slowly accelerating C-metric", proposed in [62]. It has been remarked that various solutions can appear depending on the parameter A. For A < 1/ℓ, it has been suggested that a single black hole can be built in the AdS geometry space with the only horizon being that of such a black hole. For A > 1/ℓ, however, it involved two black holes separated by the acceleration horizon [62][63][64]. According to [65], a C-metric with a cosmological constant and charge in the Hong-Teo coordinates can be represented by the metric given in Eq. (11). The first law of thermodynamics for accelerating black holes with a varying conical deficit and a critical behavior has been investigated in many places including [45]. It has been extended to f ðRÞ gravity backgrounds in [53]. The mass M of the slowly accelerating f ðRÞ AdS black hole can be calculated as 3 Advances in High Energy Physics While its charge is given by Roughly, the entropy of the black hole reads as whereĥ indicates the determinant of the induced metric on the t = const: and r = r + hypersurface. The quantity ε ab is a normal tensor verifying ε ab ε ab = −2. In Figure 1, we illustrate the variation of such an entropy quantity in terms of the accelerating parameter A and the horizon radius r + .
The left side of Figure 1, associated with the fixed horizon radius, shows that the entropy presents a decreasing variation in terms of the accelerating parameter A and the f ðRÞ gravity parameter η. For a fixed acceleration parameter A (right side of Figure 1), it follows that the entropy grows up within the horizon radius r + . Moreover, we observe that the entropy increases by increasing r + and blows up at ðAr + Þ 2 = 1. Following the η parameter, the entropy grows as η increases.
Roughly, the temperature of the black hole is given by It has been quite well-known that the entanglement entropy (EE) is considered as a good approach to measure the amount of quantum information of a bipartite system. One way to quantify this information is to compute the von-Neumann entropy of a bipartite system where the system is divided into two parts. For this proposal, we first consider a time slice located at the AdS spacetime. Then, we calculate the area of the minimal surface γ A . This can be parameterized by r = rðθÞ given by a time slice t = 0 in the line-element appearing in Eq. (11). In this case, the entanglement entropy can be expressed as where G is the Newton's constant [6,7]. Using Eq. (26) and the metric presented in Eq. (11), the holographic entanglement entropy (HEE) for charged accelerated black holes in f ðRÞ gravity can be examined. The area, associated with the minimal hyper-surface in the presence of a charged accelerating AdS black hole in the bulk, is computed using the following relation In fact, this surface area should be minimized. A Lagrangian, associated with such an area, can be worked out to give the equation of motion. Indeed, the latter reads as

Advances in High Energy Physics
A priori, they are many ways to solve this equation. However, the analytic one seems to be hard. For such a reason, we could use numerical procedures to get rðθÞ. The Wolfram Mathematica program used to support the findings of this study is available from the corresponding author upon request.
It follows that the values of ΔS A decrease as a function of the acceleration parameter when the charged accelerating AdS black hole is located in the bulk. At this level, one may give some discussing points. The first one concerns the similarity with the behaviors reported in the study of the entanglement entropy within the framework of the relativistic quantum information theory [51,52]. The second point is that the variation of such a quantity is affected by the values of η. Two regions associated with such behaviors can appear. For the small values of the accelerating parameter A, one can note that the entanglement entropy is large as the value of η is small. While in the large values of A, the behavior changes contrarily with respect to the first region. These regions meet at a transition-like point, where all the graphs intersect.
Before going ahead, a brief comment on the behavior of the entropy given by Eq. (24), and the holographic entanglement entropy given by Eq.(26) is needed. Indeed, both quantities share the same behavior apparently. Indeed, they both decrease when the accelerating parameter A increase. For the η parameter, however, the transition point observed in the holographic picture is absent in the entropy one. Since the holographic entanglement entropy is not dual to the black hole entropy, they should not necessarily present the same behavior. This difference observed here commensurate with [20,21], where the authors demonstrate that the Maxwell law observed in ðT, SÞ plane is not verified in ðT, ΔS A Þ one. The apparently similitude, observed in this case, is due to the fact that HEE depends on mass directly.

Two-Point Correlation Function of Charged Accelerating Black Holes in fðRÞ Gravity Background
Having investigated the behavior of the holographic entanglement entropy as a function of the accelerating parameter A and the parameter η, we move to discuss two-point correlation functions. In particular, we will show that such a local observable exhibits the same scheme found in the previous section.
To show that, we first give a concise review of the Maldacena correspondence in order to introduce the time two-point correlation functions under the saddle-point approximation and in the large limit of Δ as [47].
It is noted that Δ stands for the conformal dimension of the scalar operator O in the dual field theory. L is the length of the bulk geodesic between the two points ðt 0 , x i Þ and ðt 0 , x j Þ on the AdS boundary. Using spacetime symmetry arguments of the associated black hole, we can redefine x i as θ with the boundary θ 0 to parameterize the trajectory. In this way, the proper length takes the following form

Advances in High Energy Physics
where one has used r ′ = dr/dθ. Treating L as Lagrangian and θ as time, the equation of motion associated with rðθÞ reads as Using the same boundary conditions used in the previous section r ′ ðθÞ = 0 and r = r 0 at θ = 0 and r = −1/ðA cos θ 0 Þ at θ = θ 0 , we can solve such an equation by choosing the same values of θ 0 with the same UV cutoff in the dual field theory. Labeling the regularized two-point correlation functions as Δ L A = L − L 0 , where L 0 is the geodesic length in Rindler-AdS under the same boundary region obtained with the help of Eq. (29), we present the variation of the function of ΔL A in terms of the acceleration parameter A. The associated computation is illustrated in Figure 3. It is observed, from this figure, that the same behavior is held as the HEE picture.

Discussions and Concluding Remarks
In this paper, the holographic tools including entanglement entropy and two-point correlation functions have been investigated for accelerating observers in four-dimensional f ðRÞ gravity backgrounds. In particular, we have considered a C-metric with a cosmological constant in the Hong-Teo coordinates to evaluate the minimal surface area associated with HEE and two-point correlation functions of a charged single accelerated AdS black hole in four dimensions.
Concretely, we have dealt with the acceleration parameter less than the inverse of the cosmological length ℓ, where a single black hole appears in the AdS geometry with the only horizon being that of a black hole. Due to the appearance of a communicating horizon, there is no possibility for a uniformly accelerated observer to get information about space-time behaviours. A loss of information takes place generating an entanglement degradation which implies that, in a space-time with horizons, one might expect some kinds of information loss puzzle [2,54]. For such a reason, various activities have been made to figure out how the holographic entanglement entropy and two-point correlation functions behave as a function of the accelerating parameter A and the parameter η controlling f ðRÞ gravity effects. It has been observed that the holographic entanglement entropy and two-point correlation functions decrease by increasing the acceleration parameters of such black hole solutions. This indicates that the holographic entanglement entropy follows the universal behavior of entanglement entropy for accelerating observers [51,52].
Moreover, we have investigated the effects of the f ðRÞ gravity parameter η on such nonlocal observers. Rigorously, the holographic quantities including HEE and two-point correla-tion functions exhibit nontrivial behaviors. It has been observed a transition point where the behavior of the holographic tools changes. We have found two regions intersecting at such a transition point. These two regions correspond to slow and fast accelerating black holes in our nomination, retrospectively. In the first region, HEE and two-point correlation functions decrease by increasing the η parameter. However, the second one is associated with a reversed behavioral situation. A comparison between the nonlocal observables and the entropy has been investigated. The absence of the transition point in the entropy picture provides a counterexample showing that both entropy and nonlocal observable exhibit a similar behavior.
This work comes up with some open questions. It will be interesting to extend the calculation for A > 1/ℓ with two black holes separated by an acceleration horizon. Motivated by string theory and related topics, higher dimensional solutions could be developed and investigated in the presence of nontrivial contributions including dark energy effects. Moreover, the thermodynamical behavior within the nonlocal observable quantities could be considered as interesting investigated approaches. We leave these questions for future works.

Appendix
The complete form of the event horizon radius is  Advances in High Energy Physics

Data Availability
Since this work is theoretical, there is no data used to support the findings of this study. The Mathematica notebooks associated with this study are available from the corresponding author upon request.

Conflicts of Interest
The author declares that they have no conflicts of interest.