Horizon Thermodynamics in D-Dimensional f(R) Black Hole

We consider whether the new horizon-first law works in higher-dimensional f(R) theory. We firstly obtain the general formulas to calculate the entropy and the energy of a general spherically-symmetric black hole in D-dimensional f(R) theory. For applications, we compute the entropies and the energies of some black hokes in some interesting higher-dimensional f(R) theories.


Introduction
Since Bekenstein's and Hawking's work [1,2], it is convinced that there may be a deep relation between the gravitational field equations and the laws of thermodynamics. Like in thermodynamics, four laws of black hole dynamics were found in [3]. The field equations of general relativity in its tensorial form can be derived by applying the Clausius relation δQ = TδS on the horizon of spacetime, here δQ is the energy flux across the horizon and δS and T are the change in the entropy and the Unruh temperature seen by an accelerating observer just inside the horizon [4]. From Einstein equations, one can obtain the thermal entropy density of spacetime without assuming the temperature or the horizon [5,6]. It was shown that for a generalized gravity theory, the field equations are equivalent to the first law of thermodynamics [7]. This programme was also applied to other modified gravity theories: such as f (R) theory [8,9], and scalar-Gauss-Bonnet gravity [10]. It was shown, however, that the Bekenstein-Hawking entropy depends not only on the black hole parameter, but also on the coupling which induces Lorentz violation [11].
For a spherically-symmetric spacetime, Einstein's field equations can be written in the form of thermodynamic identity (called the horizon-first law): dE = TdS − PdV [12]. This framework of horizon thermodynamics has also been extended to other theories of gravity [13,14] and the non-spherically-symmetric cases [15]. The horizon-first law, however, has two shortcomings: (a) the thermodynamic variables are vague in the original derivation and require further determination, (b) both S and V are functions of only r + , so does the horizon-first law, which makes the terms 'heat' and 'work' confused [16]. To avoid these two problems, a new horizon-first law was proposed in [16], where the temperature T and the pressure P are independent thermodynamic quantities and the entropy and free energy are derived concepts, while the horizon-first law can be restored by the Legendre

The New Horizon-First Law and Its Application in f (R) Theory
Inspired by the radial Einstein equation on the horizon of Schwarzschild black hole, it is reasonable to suggest that the radial field equation of a gravitational theory under consideration takes the following form [16] where C and D are functions of the radius of black hole, r + , in general they depend on the gravitational theory one considered. The temperature T in (1) is identified from thermal quantum field theory, which is independent of any gravitational field equations [16]. According to the conjecture proposed in [6], the pressure in (1) is identified as the ( r r ) component of the matter stress-energy, it also does not fall back on any gravitational field equations. Considering a virtual displacements δr + and varying the Equation (1), then multiplying the volume of black hole V(r + ), yields [16] comparing with the thermodynamical identity δG = −SδT + VδP, where G can be identified as the Gibbs free energy which is given by [16] G = V(r + )D (r + ) dr + + T V(r + )C (r + )dr + and S is identified as the entropy which is [16] S = V (r + )C(r + )dr + .
Using the degenerate Legendre transformation as that in thermodynamics, the energy E is defined as E = G + TS − PV which can be easily got [18] This procedure was firstly investigated for Einstein gravity and Lovelock gravity which only give rise to second-order field equation [16] and was also applied to f (R) gravity with a general static spherically-symmetric black hole in f (R) gravity where W(r) and N(r) are general functions of the coordinate r and the event horizon is local at the largest positive root of N(r + ) = 0 with N (r + ) = 0, the entropy in this case is given by [19] S = (2πr + F + πr 2 + F )dr + = πr 2 + F, where F = d f dR . The energy is found to be [19] For W(r) = N(r), Equation (8) reduces to the result obtained in [18] which is consistent with the expression obtained in [21] and can be derived by using the unified first law of black hole dynamics [29]; Equation (7) is consistent with the results derived by using the Wald entropy formula or the Euclidean semiclassical approach [30][31][32]. In the next section, we will consider the new horizon-first law in D-dimensional f (R) Theory with a general static spherically-symmetric black hole.

The Entropy and Energy of D-Dimensional f (R) Black Hole
In this section, we turn our attention to discussing whether the new horizon-first law still holds in the D-dimensional f (R) theory. Considering a general spherically-symmetric and static D-dimensional black hole in f (R) theory, its geometry is given by in which dΩ 2 D−2 represents the D − 2-dimensional unit spherical line element. For the metric (9), the surface gravity takes the form [33]: κ K = W (r + )N (r + )/2, giving the temperature of the black hole as The action of D-dimensional f (R) gravity with source is represented by where k 2 = 8π and D ≥ 3. Here we take the units G = c =h = 1. f (R) is a function of the Ricci scalar R and L m is the matter Lagrangian. Physically f (R) theory must fulfil two stability conditions [34]: (a) no ghosts, d f /dR > 0; and (b) no tachyons, d 2 f /dR 2 > 0 [35]. Variation of the action (11) with respect to metric provides the gravitational field equations where T µν = −2 √ −g δL m δg µν the energy-momentum tensor of the matter. We define the stress-energy tensor of the effective curvature fluid as T ν µ which is given by where 2 = ∇ λ ∇ λ . Assuming the metric (9), we derive after some calculations by using of the relations 2F = 1 √ −g ∂ µ [ √ −gg µν ∂ ν F] the ( 1 1 ) components of the Einstein tensor and the effective curvature fluid respectively and where the prime stands for the derivative with respected to r. Taking the trace of Equation (12), yields the relation where T ν ν is the trace of the energy-momentum tensor. Substituting Equations (14), (15) and T r r = P into Equation (12), yields Thinking of N(r + ) = 0 and the temperature (10) at the horizon, Equation (17) reduces to Comparing Equations (18) and (1), we then get and The volume V of the black hole in D-dimensional spacetime takes the form [36] V(r + ) = 2π Use the relation Substituting Equations (22) and (20) into the expression (4), the entropy of black hole (9) in D-dimensional f (R) gravity is Inserting Equations (22) and (19) into Equation (5), then we obtain the energy of black hole (9) in D-dimensional f (R) theory as Equations (23) and (24) are the main results obtained in this work, they can be used to calculate the entropy and the energy of a specific black hole in a specific D-dimensional f (R) gravity. For D = 4, Equations (23) and (24) recover Equations (7) and (8). Fixing f (R) = R and W = N, one expects the results to go back to the framework of higher-dimensional Einstein's gravity, see Equations (27) and (29) in the next section.

Applications
In this section, we will illustrate the procedure to calculate the entropy and the energy for black holes in a certain f (R) theory by using Equations (23) and (24). These models have solutions with constant Ricci curvature (such as a Schwarzschild or a Schwarzschild-de Sitter solution) or solutions with non-constant Ricci curvature.

The Constant Ricci Curvature Case
We start with the simplest but important case, F = 1, which implies f = R − 2Λ where −2Λ is an integration constant to be regarded as the cosmological constant. This model has a Schwarzschild or a Schwarzschild-de/anti de Sitter black hole solution [37,38] where M is the mass of black hole. The solution is Schwarzschild-de/anti de Sitter black hole solution for D > 3 and it is the non-rotating BTZ black hole for D = 3. The constant curvature R 0 from Equation (25) is given by From Equation (23), the entropy is found to be It reduces to S = πr + /2 for a non-rotating BTZ black hole and returns to the standard results for D = 4. The energy of the black hole is obtained from Equation (24) as where −Λr 2 + = M for D = 3 has been used at the horizon. For M = 0, it reduces to results presented in [12]. For D > 3, it reads where we used N(r + ) = 0 for D > 3 at the horizon. For 4-Dimensional Einstein's gravity, Equations (27) and (29) give S = πr 2 + = A/4 and E = M, respectively. The nonnegativity of the energy, gives new constrains on the parameter:

The Non-Constant Ricci Curvature Case
We apply the same procedure for black hole solutions with non-constant curvature which are more interesting. We consider two types of f (R) theories: (a) F is a linear function of r, and (b) F is a power law function of r.

F(r) = 1 + αr
In this case, F is a linear function of r with α a non-zero constant. W(r) and N(r) in three-dimensional spacetime are given by [37] W(r) = N(r) = C 2 r 2 + C 1 αr − 1 2 − C 1 α 2 r 2 ln 1 + 1 Function f (R(r)) reads where C i are integration constants with C 1 related to the mass of the central object and C 2 identified as the cosmological constant. The Ricci scalar R evolves as Then, the entropy formula (23) gives gives limit on parameter: α ≥ −1/r + from S ≥ 0. The energy of the black hole is obtained from Equation (24) as where N(r + ) = 0 was used. E ≥ 0 gives a new constraint on the parameter: C 1 ≥ 0. For D = 4 spacetime, W(r) and N(r) take the forms W(r) = N(r) = C 2 r 2 + 1 2 where C 2 is related to the cosmological constant. We note that Equation (35) is different from Equation (27) in [37]. Function f (R(r)) is given by with the Ricci scalar From Equation (23), the entropy of the black hole reads The energy of the black hole is obtained from Equation (24) as where we used N(r + ) = 0. For C 1 = 0, Equation (39) reduces to the result in [18]. To guarantee the nonnegativity of the entropy and the energy, we must have new constraints on the parameters: α ≥ −1/r + and C 1 ≥ 1/6α.

F = αr a
We now consider a power-law form for F(r), i.e., F = αr a , with constants a and α. In this case, the W(r) and N(r) in (9) were found to be [37] and where C 1 and C 2 are the integration constants. It returns to the Schwarzschild-de/anti de Sitter solutions for a = 0 and α = 1. Function f (R(r)) and the Ricci scalar R take the forms, respectively Note that although α and a are two arbitrary constants, a must satisfy a = 2 − D, 1 ± √ D − 1. From Equation (23) the entropy for this type black hole is The energy of the black hole is obtained from Equation (24) as where and If taking a = 0 and α = 1, it is back to Einstein's gravity and Equation (44) reduces to Equation ( 2(1−a) and a = 0. f (R) is a constant for a = 3 and it is un-physical for a = 1. The entropy (44) and the energy (45) respectively reduce to and 4a+2 . The nonnegativity of the entropy gives constraints on the parameters: C 1 (a + 1) ≤ 0, and S ≥ 0 gives α ≥ 0. For a = 1/3, we have f ∼ R 2 , S = α 2 πr 4/3 15 and E = − αC 1 6 . For D ≥ 4 and C 2 = 0, the function f (R) takes the form [37] where The entropy (44) and the energy (45) respectively reads

Discussion and Conclusions
We have discussed whether the new horizon-first law still holds in higher-dimensional f (R) gravity. We have derived the general formulas to calculate the entropy and the energy of a general spherically-symmetric and static D-dimensional black hole in f (R) theories, which can be obtained by using other methods. It gives a new method to rapidly compute the entropy and the energy of the black hole in f (R) theory. For applications, we have calculated the entropy and the energy of some black holes with constant Ricci curvature or with non-constant Ricci curvature in some interesting f (R) theory by using these formulas, the nonnegativity of the entropy and the energy give new constraints on the parameters. Except for the case discussed in [39] where F(R) = 0, it is valuable to apply this procedure to other modified gravitational theories.