Exact charged black hole solutions in D-dimensions in f(R) gravity

We consider Maxwell-f(R) gravity and obtain an exact charged black hole solution with dynamic curvature in D-dimensions. Considering a spherically symmetric metric ansatz and without specifying the form of f(R) we fnd a general black hole solution in D-dimensions. This general black hole solution can be reduced to the Reissner-Nordstr\"om (RN) black hole in D-dimensions in Einstein gravity and to the known charged black hole solutions with constant curvature in f(R) gravity. Restricting the parameters of the general solution we get polynomial solutions which reveal novel properties when compared to RN black holes. Specifcally we study the solution in (3 + 1)-dimensions in which the form of f(R) can be solved explicitly giving a dynamic curvature and compare it with the RN black hole. We also carry out a detailed study of its thermodynamics.

In this work we study exact charged black hole solutions in general D-dimensional f (R) gravity.
We consider a spherically symmetric ansatz with one unknown function and we do not specify the form of the function f (R). Solving the system of the Einstein-Maxwell equations we find the form of the f (R) function which can be resulted from non-polynomial Lagrangians giving in this way more general f (R) modified gravity theories. We show that constraining the parameters of the general solution we get polynomial solutions which reveal some interesting properties when compared with RN black holes. We then focus on the (3 + 1)-dimensions, where the form of f (R) can be solved explicitly from the polynomial solution and we discuss in details the thermodynamics of this solution studying the Bekenstein-Hawking entropy, the quasi-local energy, the Hawking temperature and the heat capacity.
The work is organized as follows. In Section II we discuss the general D-dimensional solution. In Section III we present some special solutions with constant and non-constant curvature. In Section IV we discuss solutions in various dimensions. In Section V we elaborate on (3 + 1)-dimensional solutions. Finally Section VI are our conclusions.

II. GENERAL SOLUTION IN D-DIMENSIONS (D ≥ 3)
We consider a D-dimensional action in which except the Ricci scalar R there is a general function of f (R) which is not specified, a Maxwell field and a cosmological constant Λ By variation of the above action we obtain the field equations where A a = h(r)(dt) a .
We consider a spherically symmetric metric ansatz with only one unknown function that includes three basic topologies under which the components of the Einstein equation are where f (r) ≡ f (R(r)) and is the Ricci scalar expressed by the metric function B(r) and its derivatives B (r) and B (r).
The eq. I t t − I r r = 0 gives a simple relation which leads to where c 1 and c 2 are integration constants. This relation gives where the parameter c 1 is dimensionless and c 2 have dimensions [c 2 ] = L −1 . Therefore if c 1 and c 2 are small this theory can be considered as a small perturbation of the Einstein gravity.
The relation (18) is one of the central relations in our work. If c 2 is zero then it tells us that only linear curvature modifications are present. If however c 2 is not zero, then non-linear curvature terms are present in the theory.
Besides, the t component of the electromagnetic field equation gives which leads to where q and φ 0 are integration constants.
Using the eq. (20) we can find that the parameter q is proportional to the charge Q where is the volume of the unit (D − 2)-sphere.
Substituting the expressions of f R (r) and h(r) in the Einstein equations, we are getting two independent equations with two unknown functions f (r) and B(r), which contain both D > 3 and D = 3 cases. We solve f (r) from I θ 1 θ 1 = 0, and put it back into I t t = 0, then a simple second order differential equation with respect to the metric function B(r) is obtained from which we get the general exact solution in D-dimensions (D > 3) for the metric function B(r) where c 3 and c 4 are constants of integration, while 2 F 1 and 2 F * 1 are the hypergeometric function and the Regularized generalized hypergeometric function respectively. This general solution is obtained using the metric ansatz eq. (8) without introducing any extra conditions. Then the form of f (R) is specified once the metric function B(r) is known from eq. (27). Although the explicit form of f (R) is unknown, once the known functions f (r) and R(r) are specified we can have a differential equation of f (R) the solution of which gives the form of f (R). This is a novel result and in the following we will reproduce known and new f (R) solutions. It is worth noticing that the dimension D can be any real number larger than 3, not constrained to integers. In Sec. IV, we will show that for integer D, the solution only contains polynomials and logarithmic terms.
For D = 3 we obtain a new exact charged black hole solution −2c 2 2 q 2 r 2 ln 2 r + 4c 2 2 q 2 r 2 ln r ln where Li 2 is the polylogarithm function, defined by a power series in z, which is also a Dirichlet series in s The cosmological constant Λ does not show up in the solution, but we can know if the space is flat, AdS or dS by analysing its asymptotic behaviors at spacial infinity and origin where In (2 + 1)-dimensions, near horizon solutions, asymptotically Lifshitz black hole solutions and rotating black holes with exponential form of f (R) theory have been discussed in [38]. They first gave the basic field equations as same as our equations with D = 3, but their solution is different with our solution.

III. SPECIAL SOLUTIONS
We will first consider solutions with c 2 = 0. In this case f (R) = c 1 and the eq. (26) becomes the solutions of which is and then the functions R(r) and f (r) become Then we have two cases: Non-constant R For non-constant R, using the the solutions (35) (36) we have for f (R) and then to have f (R) = c 1 R we get for the parameter Then the metric function and the curvature function become This result implies that it reduces to the Einstein Gravity R+f (R)−2Λ = (1+c 1 )R−2Λ = R−2Λ when c 1 = 0. In this case the solutions (39) become which after some parameterization are exactly the standard higher-dimensional charged black hole solutions [39] and the charged BTZ black hole solution [40] in Einstein-Maxwell theory with dynamic curvatures where q 1 and m 1 are related to the electric charge and the ADM mass of the BH, and is the AdS For constant curvature R = R 0 , we can take the trace of the Einstein equation and obtain where Maxwell electromagnetic field is traceless T = g µν T µν = 0. This relation can lead to To see this consider the solutions eq. (35). We can see that for constant curvature the parameters must satisfy q = 0 or D = 4. The former condition q = 0 leads to which hold for all dimensions D ≥ 3. Comparing with the eq. (47), we can obtain the relation between the parameter c 1 and the constant curvature R 0 These geometries are exactly the same with the Schwarzschild black hole solutions in D-dimensions and the BTZ black hole solution in 3-dimensions, Specially in 4-dimensions, the Ricci scalar is always a constant even with nonzero q, where c 1 has the same relation as in eq. (50).
Note that f (R 0 ) = c 1 , after parametrizations we have which has been studied in [34,36]. However this kind of solutions can not be distinguished with the RN black holes in Einstein Gravity, since we can always adjust the gravitational constant to make them the same.
It is worth noticing that this charged solution with constant curvature only exists in 4-dimensions while the Schwarzschild solution can be present in any higher dimensions.

IV. EXPLICIT SOLUTIONS IN VARIOUS DIMENSIONS
In this Section we will discuss the forms and the properties of the general solution (27)  We have found the solution for D = 3 eq. (28), to compare with the BTZ black hole we set q = 0 and then the metric function becomes The asymptotic behaviors of the metric function are where the leading order at r → ∞ is r 2 term, so we define its coefficient as the effective cosmological the sign of which can determine the property of spacetime to be AdS or dS or if Λ eff = 0 to be flat.
To check the possibility of flat and dS black hole, we introduce a root function the roots of which are also the roots of the metric function B(r). The derivative of Root(r) is always positive under the condition and the asymptotic behaviors of root function at r → 0 and r → ∞ are respectively where Λ eff B 0 > 0 indicates one horizon and Λ eff B 0 ≤ 0 indicates no horizon. If we want a dS black hole, there at least two horizons exist, while if we want an AdS black hole, one horizon is required. It is clear that the solution we obtained can only represent AdS black hole spacetimes (Λ < 0 and B 0 < 0) or pure dS spacetimes (Λ > 0 and B 0 > 0).
The general solution (27) contains some special functions that are not easy to analyse. However, when we solve the eq. (26) in each dimension, the solutions become much simpler, only containing polynomials and logarithmic terms. In the Appendix A we give the solutions for D = 4, 5, 6 dimensions. The solution (A1) for D = 4 has been discussed in [32] while the solutions in higher dimensions are new and they have not been studied before.
To compare with the RN black holes and understand the physical meaning of the constants of integration, we set the coefficients of the logarithmic terms to be zero and then the solutions are just polynomials. Then from the solutions (A1), (A2) and (A3) we obtain the general constraint for the parameters under which the solutions (A1), (A2) and (A3) can be reduced to simpler polynomial solutions in Compared with RN black hole, our solution (64) has more terms from 1/r to 1/r 2(D−3) while RN black hole only has two terms 1/r and 1/r 2(D−3) as mass term and charge term respectively. Another interesting difference is that the constant term in our solution is a fraction (D−3)k D−2 depending on dimension D, and this fraction can not be rescaled. Moreover, the dynamic curvature R(r) and the nonzero gravitational action f (r) can not be simplified by the transformation of coordinates.
Using relations (14) and (26) we can obtain the expressions of R(r) and f (r) and using the relation (16) we can get the exact expression of f (R), Defining an effective cosmological constant we give the functions B(r), R(r), f (r), and f (R) in the Appendix B.
In (3 + 1)-dimensions the form of f (R) can be solved explicitly so we will discuss this solution in details studying also its thermodynamics.
In the Appendix A the solution (A1) in (3 + 1)-dimensions was given from which we can have an explicit form of f (R) where the constant of integration c 5 should be set to to ensure that the only constant in the action is −2Λ. After this constraint the f (R) becomes Substituting R(r) into this expression and comparing with f (r), we finally get then the solution becomes with where Λ = Λ 1+c 1 is an effective cosmological constant and κ = κ 1+c 1 . Then we can obtain the other equivalent formF (R)F In fact we can always rescale to make 1 + c 1 = 1, then then we can adjust the parameter c 2 to change the relation between Einstein action term R and the nonlinear action term k R − 4Λ 1+c 1 . Note that when c 1 = 0 and c 2 = 0 the action (80) and the general solution will reduce to the Einstein Gravity and RN black hole, but for this brunch with c 2 = 0, the solution can never reduce to the standard RN black hole solution even when c 1 = 0 and c 2 approaches to 0. Here we give the solutions B 0 (r), R 0 (r), f 0 (r) and F 0 (r) to represent the solutions B(r) , R(r), f (r) and F (r) with In the following figures we depict the plots of B 0 (r) to show the influence of parameter c 2 . In larger c 2 has smaller radius of event horizon, which means that larger nonlinear gravitational action gives smaller black hole.
In Fig. 2, the figures are plotted with negative c 2 . It shows that for k = 1 the black hole with larger absolute value of c 2 has smaller radius of event horizon, which means that larger proportion Compared with RN black hole in (3 + 1)-dimensions, our solution has two terms related to the parameter k, which means the topology has more influence on the geometry. Besides, our solution also contains a dynamic curvature R(r) and non-zero gravitational action F (r), while the RN black hole has a constant curvature and zero gravitational action. Except the difference of the constant terms, we can rescale the parameters to make their metric functions very similar In Fig. 3, we depict the plots of the metric functions B RN (r) and B 0 (r) with different parameters m 1 to observe the changes when the nonlinear term is present. Note that, no matter how we change the parameters, the metric functions B RN (r) and B 0 (r) always differ with a constant k 2 , so we mainly plot the figures for B 0 (r). For black holes with the same charge and the cosmological constant, with the increase of m 1 (the nonlinear action part alleviates), the black holes have larger radius of event horizons and deeper depressions of geometry inside the event horizons.
Similar solution has been obtained in a recent paper [37] with an action of the form F (R) = The equations B(r ext ) = 0 and B (r ext ) = 0 give the extremal conditions where the condition 16c 2 2 q 2 1 ≤ 1 is required for k = 1 but not required for k = −1. If c 2 < 0, we have two extremal situations for k = 1 , while for k = −1 there is only one extremal situation If c 2 > 0, the black holes can never become extremal for k = 1, while there is one extremal situation for k = −1, From the extremal conditions we can see that for k = 1 the parameter c 2 is always negative, which agrees with the discussion above. It means that the geometry only permits the positive mass for k = 1 (If mass has the opposite sign with c 2 ). Besides, it shows that the condition r ext > √ 2q 1 gives positive Λ while r ext < √ 2q 1 gives negative Λ, which means that asymptotic dS spacetimes require large enough charge to make black holes extremal, while asymptotic AdS spacetimes need enough small charge for the extremal black holes to exist.
For k = −1, the cosmological constant Λ is always negative, which is the same with the RN black holes in Einstein Gravity. It means that only asymptotic AdS spacetimes can accommodate black holes with k = −1 topology. In addition, the condition r ext < √ 2q 1 gives positive c 2 while r ext > √ 2q 1 gives negative c 2 .

B. Thermodynamics
In this subsection we will study the thermodynamics of the (3 + 1)-dimensional charged black hole solution eqs. (83)-(86) under the action are given by [41,42] S(r + ) = 1 4 AF R (r + ) = πr 2 where A is the area of the black hole, and r + is the event horizon. The Bekenstein-Hawking entropy consists of two parts: the first part πr 2 + = A/4 is proportional to the area, same as in Einstein Gravity, while the second term comes from the modification of gravity which will disappear when For a given c 2 , the r + only changes depending on q or Λ. Then the derivative of entropy is S (r + ) = πr + (2 + 3c 2 r + ), where S (r + ) ≥ 0 can give the limit on the event horizon r + ≥ − 2 3c 2 , which indicates that the radius of event horizon must be larger than the critical value − 2 3c 2 . In other words, enough small black holes are not permitted in this f (R) theory.
On the other hand, if we fix q and Λ, then the r + only changes depending on c 2 . In the following figures (Λ > 0) we can see that the second law of thermodynamics will be violated by black holes with enough large radius of event horizon, which means that the parameter c 2 in this f (R) theory can not be larger than a critical value that depends on the given q and Λ.
Besides the Hawing temperature is and the heat capacity C is [43,44] We know that when c 2 , q 1 and Λ are given, the event horizon will be fixed. If only q 1 and Λ are fixed the parameter c 2 can be expressed as a function of the event horizon r + Then the Bekenstein-Hawking entropy S(r + ), the quasi-local energy E(r + ), the Hawing temperature T (r + ) and the heat capacity C(r + ) can be written as functions of q 1 and Λ S(r + ) = πr 2 In the following we will discuss the thermodynamics of a spherically symmetric solution with various values of the cosmological constants Λ.
We will first consider the case of Λ = 0 and k = 1, then from eq. (104) we have where 18c 2 2 q 2 1 ≤ 1, it is clear that c 2 must be negative − 1 In this case we have where for the Bekenstein-Hawking entropy and the quasi-local energy to be positive.
Note that as c 2 varies in the range there are two horizons, the larger one is the event horizon and the smaller one is the Cauchy horizon.
With the decrease of c 2 , Cauchy horizon grows while the event horizon shrinks until they coincide, where is only one horizon at the extremal limit From Fig. 5 we can see that for asymptotically flat spacetimes the Bekenstein-Hawking entropy S(r + ) grows with r + in quadratic form while the quasi-local energy E(r + ) rises with r + linearly.
Besides, large q 1 gives large entropy and quasi-local energy.
The Fig. 6 shows that the Hawing temperature grows from the extremal case, where T (r ext ) = 0, after reaching its maximum value, it decreases slowly to zero at infinity. From the expression of heat capacity eq. (108), we can see that for asymptotic flat spacetimes there is a divergent point r + = 2q 1 . For the black holes smaller than this critical value, the heat capacities are positive, while the larger black holes have negative capacity. We know that ordinary matter have positive capacity which means that to maintain matter in higher temperature, more energy is required. As opposed to that, negative capacity means that a black hole needs to lose energy to get higher temperature.
Positive divergence of heat capacity indicates the temperature could not get higher temperature even if the black hole absorbs much more energy. While the negative divergence of heat capacity indicates the temperature could not get higher even if the black hole loses much more energy.
Therefore the divergent points correspond to the maximum values of temperature.
and then 16Λq 2 1 ≤ 1 is required. For the roots of metric function B 0 (r), the horizon r h can be Cauchy horizon, event horizon or cosmological horizon, to confirm the extremal cases, we need to check the sign of B 0 (r h ) positive while B 0 (r ext+ ) is always negative, which means that for dS spacetimes the "−" indicates the coincide of the Cauchy horizon and the event horizon while the + indicates the coincide of event horizon and the cosmological horizon.
So we can obtain the ranges of r + and c 2 from the extremal conditions where Now we can draw the plots of the horizons and the quasi-local energy as functions of c 2 , also the Bekenstein-Hawking entropy S(r + ), the Quasi-local energy E(r + ), the Hawing temperature T (r + ) and the heat capacity C(r + ) as functions of r + . In Fig. 7, there are three horizons for l 1 < c 2 < l 2 , the largest one is the cosmological horizon, the smallest one is the Cauchy horizon and the middle one is the event horizon. When c 2 = l 1 , the Cauchy horizon and event horizon coincide, corresponding to extremal black holes. Then with the increase of c 2 , the extremal horizon splits into two horizons: the Cauchy horizon and the event horizon, and the radius of event horizon grows while the cosmological horizon shrinks and they meet each other at c 2 = l 2 . We can also see that larger charge gives larger Cauchy horizon, smaller event horizon, it makes the extremal black hole more easily happen, corresponding to smaller ranges of c 2 . On the other hand, charge almost does not have any influence on the cosmological horizon.
The quasi-local energy always rises with the increase of c 2 , and larger charge gives less quasi-local energy, which means that the electromagnetic field can extract energy from black hole and make it shrink. Charge can also make the relation E(c 2 ) to become sharper. In Fig. 8 we can see that for asymptotically de Sitter spacetimes the Bekenstein-Hawking entropy S(r + ) first increases gradually from a small value S(r ext− ) when the Cauchy horizon coincides with the event horizon, then it decreases to another small value S(r ext+ ) when the event horizon coincides with the cosmological horizon. The entropy is always non-negative within the range of the r + given in eq. (118). While the quasi-local energy E(r + ) always grows with the increase of r + .
When the event horizon coincides with the cosmological horizon it gets a peak value, which means that the energy inside the event horizon always becomes larger when the radius of the event horizon rises. Besides, for black holes with same radius of event horizons, charge brings more entropy and quasi-local energy.
This abnormal behavior of entropy indicates the violation of the second law of thermodynamics, which may be saved if we add external matter according to generalized second law (GSL) [45]. The generalized second law (GSL) states that the total generalized entropy S ≡ S + S bh never decreases [46], where S is the ordinary entropy of matter outside the black hole. In fact the second law is also violated in some f (R) models [47], even the generalized second law (GSL) [48].
On the other hand, if the second law holds, then for any points on the curves, the black holes will expand or shrink to reach the vertexes of the curves, where the maximum values of entropy can give the stable positions. For smaller black holes, they need to absorb matter to have more entropy, while larger black holes needs to expel matter to have more entropy. The Fig. 9 shows that for asymptotic de Sitter spacetimes, the Hawing temperature first grows from zero then decreases to zero which corresponds to the extremal. On the other hand, the peak of the temperature corresponds to the divergent point of heat capacity likewise. The difference is that for Λ > 0 it may rises across the zero and then decreases to zero, where the cross point indicates that the temperature changes sharply when the black hole absorbs or loses energy. Besides larger charge gives lower temperature and less capacity like the way in asymptotic flat spacetimes. Other properties are also similar.
For asymptotic AdS spacetimes, the extremal condition can be written as For k = 1, we have  In Fig. 11 we can see that for asymptotically anti-de Sitter spacetimes the Bekenstein-Hawking entropy S(r + ) and the quasi-local energy E(r + ) both grow with the increase of r + , and smaller cosmological constant gives larger entropy and quasi-local energy. The Fig. 12 shows that the temperature rises sharply from the zero and then decreases and then it also increases slowly. While the heat capacity is always negative beyond the divergent point, which means that the black hole still needs to lose energy for higher temperature. Besides, smaller cosmological constant gives higher temperature and lower heat capacity.
Concluding this section, we made an analytic study of the thermodynamics of a (3+1)-dimensional charged black hole which is resulting from an action that except the usual linear Ricci scalar term there is also a non-linear term of the Ricci scalar. For a spherically symmetric charged black hole we found that the asymptotic properties of spacetimes impose stringent constraints on the structure of the black hole giving consistency with the second law of thermodynamics. For example, if the asymptotic spacetime is dS, depending on the strength of the correction nonlinear term of the Ricci scalar, the charged black hole has a rich structure in order to give a valuable thermodynamic behaviour. We also expect to have similar behaviour if the topology is flat or hyperbolic of the charged black hole.

VI. CONCLUSIONS
In this work we obtained an exact charged black hole solution with dynamic curvature in The increase of vacuum energy is giving less Bekenstein-Hawking entropy and the quasi-local energy.
In the case of the dS spacetime we found that the Bekenstein-Hawking entropy S first increases gradually from a small value when the Cauchy horizon coincides with the event horizon, then it decreases to another small value when the event horizon coincides with the cosmological horizon.