Universal thermodynamics in different gravity theories: Modified entropy on the horizons

The paper deals with universal thermodynamics for FRW model of the universe bounded by apparent (or event) horizon. Assuming Hawking temperature on the horizon, the unified first law is examined on the horizon for different gravity theories. The results show that equilibrium configuration is preserved with a modification to Bekenstein entropy on the horizon.

Hawking temperature and Bekenstein entropy respectively as on the apparent horizon with R A , the radius of the apparent horizon. Further, they have shown the equivalence between the thermodynamical laws and modified Einstein field equations in Gauss-Bonnet gravity and more general Lovelock gravity. Subsequently Cai et al. [18][19][20] have extensively studied unified first law in FRW universe not only for Einstein gravity but also in Lovelock gravity, Scalar-Tensor theory [18] and Brane-world scenario [20]. In this context, Eling et al. [22] have shown for f (R) gravity theory that there should be entropy production term in the Clausius relation and it can be associated to shear viscosity of the horizon in pure Einstein gravity. Very recently, thermodynamical laws have been studied [23,24] in f (R) gravity as well as in generalized f(R) gravity with a modified version of the entropy of the horizon. In the present work, we have modified the horizon entropy suitably so that Clausius relation is automatically satisfied.
We start with homogeneous and isotropic FRW metric as where R = ar is the area radius, h ab = diag(−1, a 2 1−kr 2 ) is the metric of 2-space (x 0 = t, x 1 = r) and k = 0, ±1 denotes the curvature scalar. The above FRW metric can be written in double-null form as [18] Here, are future pointing null vectors. The trapping horizon (denoted by R T ) is defined as ∂ + R| R=RT = 0, which gives The surface gravity is defined as so for any horizon (with area radius R h ) it can be written as and it becomes, for apparent horizon with ǫ =Ṙ A 2HRA . Note that if we assume ǫ < 1, then κ is negative and hence the apparent horizon coincides with inner trapping horizon [16,18,25] (outer trapping horizon is with positive surface gravity). The Misner-Sharp energy [16,25,26] is defined as This is the total energy inside a sphere of radius R. Note that it is purely a geometric quantity and is related to the structure of the space-time as well as to the Einstein's equations [18]. For the present FRW model of the universe bounded by the apparent horizon the above expression for the energy simplifies to (10)

f(R)-gravity
In f (R) gravity, the modified Einstein-Hilbert action can be written as (in Jordan frame) [4] S = 1 16πG with S m as the matter action. Now, variation of S with respect to the metric tensor g µν gives the modified field equations in f (R) gravity as where T ν µ = diag(−ρ, p, p, p) is the energy-momentum tensor for the matter field in the form of perfect fluid. In particular for viable f (R)-gravity theory if we take then the explicit form of the modified field equations for FRW metric are given by with ρ t = ρ + ρ e and p t = p + p e . The effective energy density ρ e and effective pressure p e due to the curvature contribution has the expressions ρ e + p e = 1 8πG where R = 6(Ḣ + 2H 2 + k a 2 ) is the curvature scalar and F 1 = dF dR . The energy conservation relations arė So the effective pressure and energy density also satisfies the conservation relatioṅ Following the method proposed by Cai [18], we shall derive an expression for entropy associated with the apparent horizon of a FRW universe described by the above modified Friedman equations (i.e., Eqs. (14) and (15)). According to Refs. [16][17][18][19][20], the energy supply vector ψ and the work density W are defined as For the present model the explicit form of these quantities are and Note that only the pure matter energy supply Aψ m (after projecting on the apparent horizon) gives the heat flow δQ in the Clausius relation δQ = T dS, where A = 4πR 2 is the surface area of a sphere of radius R.
Thus according to Hayward [16], the (0,0) component of (modified) Einstein equations (i.e., Eq. (14)) can be written as the unified first law where V = 4 3 πR 3 is the volume of the sphere of radius R. Now, using the double null vectors ∂ ± as the basis, any vector ξ tangential to the apparent horizon surface can be written as As by definition the trapping horizon is characterized by so on the marginal sphere, i.e., In the present model R T coincides with R A and Moreover using (r, t) co-ordinates, ξ can be written as [18] Now projecting the unified first law (Eq. (24)) along ξ, the true first law of thermodynamics of the apparent horizon is obtained as [18,20] dE, ξ = κ 8πG dA, ξ + W dV, ξ Note that the pure matter energy supply Aψ m when projected on the apparent horizon gives the heat flow δQ in the Clausius relation δQ = T dS. Hence from Eq. (29) we have Using Eqs. (16), (17), (22) and (23), we obtain (after a simple algebra) As the Hawking temperature on the apparent horizon is given by so the above equation can be written as Hence comparing with Clausius relation δQ = T dS and integrating, we have the entropy on the apparent horizon Thus the entropy on the apparent horizon differs from Bekenstein entropy by a correction term (given in the form of integral on the right hand side of the Eq. (34)).
As light rays move along the radial direction i.e., normal to the surface of the event horizon and we have ∂ξ ± = dt ∓ adr, one form along the normal direction, so ∂ ± = − √ 2(∂ t ∓ 1 a ∂ r ) may be chosen along the tangential direction to the surface of the event horizon. Thus for event horizon, we choose as the tangential vector to the surface of the event horizon. Now, using the expression for κ from Eq. (7) and proceeding as before, the entropy on the event horizon turns out to be Here also the leading term for entropy is the usual Bekenstein entropy. Further, if we consider the conformal transformatioñ where the scalar field φ is defined as then the action (11) (in Einstein frame) now becomes [9,27] where∇ a is the covariant derivative compatible withg ab and V (φ) is the effective potential defined as Now varying the above action (39) with respect tog ab and φ we obtain the Einstein equations (of f(R)-gravity in Einstein frame) and the evolution equation for φ as So for the present FRW model the explicit form of the field equations are andφ Thus we have Thus proceeding as above, the entropy on the apparent and event horizon are respectively given by and Note that the scalar field φ in Einstein frame corresponds to a representative form of Ricci curvature in Jordan frame. In our scenario, the Einstein frame is the physical frame which gives self gravity of the scalar field's effective potential V (φ).

Scalar-Tensor Theory
In scalar tensor theory of gravity, using Jordan frame the Lagrangian is given by [28] where f (φ) is an arbitrary function of the scalar field φ having potential V (φ), L m is the Lagrangian for the matter fields in the universe.
Now varying the action corresponding to the Lagrangian (49) with respect to the dynamical variables g µν and φ the equation of motion are and where T m µν is the energy momentum tensor of the matter distribution. Hence for FRW model, the explicit form of the equation (50) and (51) are given by andφ Now choosing f (φ) = 1 + F (φ),the field equation (53) can be rewritten aṡ Hence we have Considering ξ as given by equation (28) (for apparent horizon) or by equation (35) (for event horizon) and proceeding in the same way as before, for the validity of the unified first law, the expression of the entropy on the horizon (apparent / event ) is given by and (58) Using similar conformal transformation as in f(R)-gravity we can write down the expressions of the entropy on the horizons in Einstein frame of scalar tensor theory.

Einstein Gauss-Bonnet gravity
In Einstein Gauss-Bonnet gravity, the action in (3+1) dimensions can be written as where α, the coupling parameter has the dimension of (length) 2 and I m is the matter action. Now varying the action I over the metric tensor g µν , we have the equations of motion: G µν − αH µν = T µν , where H µν = 4R µλ R λ ν + 4R ρσ R µρνσ − 2RR µν − 2R ρσλ µ R νρσλ + 1 2 g µν G is the Lovelock tensor. Hence for the metric given in equation (2), the nonvanishing components of the modified Einstein's equations are and Hereα is the Gauss-Bonnet coupling parameter which is a function of α. Now from equation (60), we have so for this modified gravity theory the expressions for the entropy on the horizon (apparent / event) is given by Gα ln(R A ), for apparent horizon (62) and Thus in the present work, we have considered universal thermodynamics for three different gravity theories (namely f(R)-gravity, Scalar-tensor theory and Einstein Gauss-Bonnet gravity) for FRW model of the universe bounded by apparent/event horizon. Assuming the temperature on the horizon as Hawking temperature we have examined the validity of the unified first law and it turns out that the entropy on the horizon is no longer the Bekenstein entropy, rather there are correction terms in integral form. An interesting result is obtained for Einstein-Gauss-Bonnet gravity. In this modified gravity theory the entropy of the apparent horizon achieves a logarithmic correction to Bekenstein entropy. This result is not trivial. One may get the similar result in loop quantum gravity and also in the holographic description (one of the promising descriptions of quantum general relativity) of entropic cosmology. Infact, for a cosmological model involving two holographic screens the universe can arrive at thermal equilibrium only after taking into account of this logarithmic correction [29,30] . Therefore, we conclude that Universal thermodynamics in different gravity theories corresponds to equilibrium configuration-there is no need of choosing any entropy production term, instead the entropy on the horizon is non-Bekenstein in form.