A Study of Universal Thermodynamics in Massive Gravity: Modified Entropy on the Horizons
aa r X i v : . [ g r- q c ] F e b A Study of Universal Thermodynamics in Massive Gravity: Modified Entropy onthe Horizons
Subhajit Saha a , Saugata Mitra b , and Subenoy Chakraborty c Department of Mathematics, Jadavpur University, Kolkata 700032, West Bengal, India.
Universal thermodynamics for FRW model of the Universe bounded by apparent/eventhorizon has been considered for massive gravity theory. Assuming Hawking temperatureand using the unified first law of thermodynamics on the horizon, modified entropy on thehorizon has been determined. For simple perfect fluid with constant equation of state,generalized second law of thermodynamics and thermodynamical equilibrium have beenexamined on both the horizons.Keywords: Massive gravity theory, Modified entropy, Generalized second law of ther-modynamics, Thermodynamical equilibriumPACS Numbers: 04.50.Kd, 98.80.-k, 95.30.Sf a [email protected] b [email protected] c [email protected] I. INTRODUCTION
Recent astronomical data from Type Ia Supernovae (SNe Ia) [1], Cosmic Microwave RadiationBackground (CMB) [2] and Large Scale Structure (LSS) [3] provides ample evidences that the presentUniverse is undergoing an expansion which is rather accelerated. The cause for this acceleration isstill debated. A group of cosmologists have been trying to incorporate this late time acceleration intostandard cosmology by the introduction of an exotic matter which has been dubbed ”Dark energy”(DE). This hypothetical matter is thought to have a huge negative pressure, thus causing the Universeto accelerate. However, in spite of extensive research, the nature of DE is still a mystery.Another group of cosmologists are of the opinion of a modified theory of gravity − a modificationof Einstein’s general theory of relativity. A common and widely used modified gravity theory is f ( R ) − gravity, where the Lagrangian density R (Ricci scalar) in the Einstein Hilbert action is replacedby an arbitrary function of R , i.e., f ( R ) (see [4] and the references therein). One can find manyother modified gravity theories (higher dimensional theories as well) in the literature. These modifiedtheories [5–10] are considered as gravitational alternatives for DE and might serve as dark matter [11].In the present work, we consider massive gravity theory as a modified theory of gravity and examinethe thermodynamical behaviour both at the apparent and event horizons. The paper is organizedas follows. Section II describes the basic features of universal thermodynamics. The basic equationsin massive gravity theory are presented in Section III. Thermodynamics in massive gravity has beendiscussed in Section IV. Finally, a brief discussion and final comments are given in Section V. II. UNIVERSAL THERMODYNAMICS: BASIC FEATURES
A lot of research in recent years has been carried out in Universal thermodynamics, mostly withan apparent horizon as the boundary. In 2006, Wang et al. [12] made a comparative study of the twohorizons (apparent and event) by examining the validity of the thermodynamical laws for DE fluidsand concluded that the Universe bounded by an apparent horizon is a Bekenstein system whereas acosmological event horizon is unphysical from the thermodynamical point of view. However, it hasbeen shown [13] that the generalized second law of thermodynamics holds (in any gravity theory) withsome reasonable restrictions for Universe bounded by an event horizon under the assumption that thefirst law holds. Further, a modified form [14] of the Hawking temperature has been identified recentlyusing which, it has been possible to show [15] the validity of both the first and the second laws ofthermodynamics for Universe bounded by an event horizon and the results obtained are independentof the fluid taken.According to thermodynamical concepts, the entropy of an isolated macroscopic physical systemsnever decrease because such systems always evolve toward thermodynamic equilibrium, a state withmaximum entropy (consistent with the constraints imposed on the system). Thus, for a universefilled with a fluid and bounded by a horizon, the (generalized) second law of thermodynamics and thethermodynamical equilibrium respectively take the forms [16, 17]˙ S h + ˙ S fh ≥ and ¨ S h + ¨ S fh < , (1)where S h and S fh are the entropies of the horizon and the fluid within it, respectively. In order todetermine ˙ S h , we shall use the Clausius relation T h dS h = δQ h = − dE h (2)and ˙ S fh can be obtained from the Gibb’s relation [13, 18] which is given by T f dS fh = dE f + pdV h , (3)where E h is the energy flow across the horizon, E f = ρV h is the total energy of the fluid bounded thehorizon, V h = πR h is the volume of the fluid and ( T h , T f ) are the temperatures of the horizon andfluid inside it, respectively.On the other hand, in the context of universal thermodynamics, the thermodynamical aspectsof dynamical black hole (BH) [19, 20] was studied by Hayward. THe concept of trapping horizonwas introduced in 4D Einstein gravity for non-stationary spherically symmetric spacetimes. As aresult, the Einstein field equations are equivalent to the unified first law (UFL). However, the first lawof thermodynamics can be derived by projecting the UFL along any tangential direction ( ξ ) to thetrapping horizon [21–23] and the Clausius relation of the dynamical BH is written as h Aψ, ξ i = κ πG h dA, ξ i , (4)where A is the area of the horizon and the energy flux ψ is termed as energy supply vector.Further, in view of universal thermodynamics, our Universe is assumed to be a non-stationarygravitational system while from the cosmological aspect, the homogeneous and isotropic FRW Universemay be considered as dynamical spherically symmetric spacetime. As a result, there is only innertrapping horizon which coincides with the apparent horizon. So it would be interesting to havethermodynamical analysis using UFL. A first step along this line was taken in 2009 by Cai and Kim[24]. They were able to derive the Friedmann equations with arbitrary spatial curvature startingwith the fundamental relation δQ = T dS at the apparent horizon of the FRW Universe. They haveconsidered Hawking temperature ( T H ) and Bekenstein entropy ( S B ) on the apparent horizon as T H = 12 πR A , S B = πR A G , (5)with R A as the radius of the apparent horizon. Moreover, they were able to show the equivalencebetween the thermodynamical laws and modified Einstein equations in Gauss-Bonnet gravity andmore general Lovelock gravity. Then Cai and others [21–23] examined the UFL in the background ofmodified gravity theories, namely Lovelock gravity, scalar-tensor theory [21] and brane-world scenario[23]. However, in f ( R ) gravity theory, one needs entropy production term [25] for the fulfilment ofClausius relation. Subsequently, thermodynamical laws have been studied [26, 27] in f ( R ) (generalized f ( R )) gravity with a modified version of the horizon entropy. Very recently, we have modified horizonentropy [28] in such a manner that Clausius relation is automatically satisfied. The present work isan extension of that work in massive gravity theory.The homogeneous and isotropic FRW model is described by its line element as ds = − dt + a ( t )1 − kr dr + R d Ω = h ab dx a dx b + R d Ω , (6)where R = ar is the area radius, h ab = diag ( − , a − kr ) is the metric on 2-space ( x = t, x = r ) and k = 0 , ± ξ ± ), the above FRW lineelement takes the form [21] ds = − dξ + dξ − + R d Ω , (7)where ∂ ± = ∂∂ξ ± = −√ ∂∂t ∓ √ − κr a ∂∂r ! (8)are future pointing null vectors.The trapping horizon (denoted by R T ) is defined as ∂ + R | R = R T = 0, i.e., R T = 1 q H + ka = R A (9)is the radius of the trapping horizon.The surface gravity is defined as κ = 12 √− h ∂ a ( √− hh ab ∂ b R ) , (10)so for any horizon having radius R h , we have κ h = − (cid:18) R h R A (cid:19) − ˙ R A HR A R h , (11)and κ A = − − ǫR A , (12)with ǫ = ˙ R A HR A is the surface gravity at the apparent horizon.In most of the modified gravity theories, the Einstein field equations in FRW model can be writtenin the form of modified Friedmann equations as H + ka = 8 πG ρ t (13)and ˙ H − ka = − πG ( ρ t + p t ) , (14)where ρ t = ρ + ρ e and p t = p + p e are the total energy density and the thermodynamic pressure, ( ρ , p ) are the corresponding quantities for the matter distribution while ( ρ e , p e ) are termed as effectivequantities due to the curvature (or other) contributions.According to Cai [21], the energy supply vector ψ and the work density W are defined as [19–23] ψ a = T ba ∂ b R + W ∂ a R , W = − T ab h ab . (15)Now, for the above modified Friedmann equations (Eqs. (13) and (14)), the explicit form of W and ψ are W = 12 ( ρ t − p t ) = 12 ( ρ − p ) + 12 ( ρ e − p e )= W m + W e (16)and ψ = ψ m + ψ e = (cid:26) −
12 ( ρ + p ) HRdt + 12 ( ρ + p ) adr (cid:27) + (cid:26) −
12 ( ρ e + p e ) HRdt + 12 ( ρ e + p e ) adr (cid:27) . (17)It should be noted that only the pure matter energy supply Aψ m will give the heat flow δQ in theClausius relation when projected on the horizon. Also the (0,0) component of the modified Einsteinequations (i.e., Eq. (13)) can be written as the unified first law [19] dE = Aψ + W dV (18)with V = πR as the volume of the sphere of radius R . Further, any vector ξ tangential to theapparent horizon can be expressed in terms of the null vectors ∂ ± as ξ = ξ + ∂ + + ξ − ∂ − . (19)As the trapping horizon is characterized by ∂ + R T = 0, so on the marginal sphere, ξ ( ∂ + R T ) = 0, whichgives ξ + ξ − = − ∂ − ∂ + R T ∂ + ∂ + R T . (20)As for the present FRW model R T = R A , so we have ∂ − ∂ + R A = 4 R A (1 − ǫ ) , ∂ + ∂ + R A = − ǫR A . (21)Hence in ( r , t ) coordinates, the tangent vector ξ can be written as [21] ξ = ∂∂t − (1 − ǫ ) H r ∂∂r . (22)Thus projecting the above UFL (i.e., Eq. (18)) along ξ , the true first law of thermodynamics at theapparent horizon takes the form [21–23] h dE, ξ i = κ πG h dA, ξ i + h W dV, ξ i . (23)As heat flow δQ corresponds to pure matter energy supply Aψ m , projecting on the apparent horizonso form Eq. (23), we obtain [28] δQ = h Aψ m , ξ i = κ A πG h dA, ξ i − h Aψ e , ξ i . (24)Now using Eqs. (17) and (12), the explicit form of δQ is δQ = − ǫ (1 − ǫ ) G HR A + A (1 − ǫ ) HR A ( ρ e + p e ) (25)with Hawking temperature on the apparent horizon as T A = κ A π = (1 − ǫ )2 πR A . (26)Hence Eq. (25) can be rewritten as [28] δQ = h Aψ m , ξ i = T A h πR A G dR A − π HR A ( ρ e + p e ) dt, ξ i . (27)On comparison with Clausius relation δQ = T dS , the differential form of the entropy on the apparenthorizon takes the form [28] dS A = 2 πR A G dR A − π HR A ( ρ e + p e ) dt (28) i.e., S A = A A G − π Z HR A ( ρ e + p e ) dt. (29)This shows that the entropy on the apparent horizon is the usual Bekenstein entropy with a correctionterm (in integral form).For the event horizon, as dξ ± = dt ∓ adr , one form along the normal direction, so the tangentvector to the event horizon can be taken as ∂ ± = −√ ∂ t ∓ a dr ) , (30)i.e., one can choose ξ = ∂∂ t − a ∂∂ r as the tangential vector to the surface of the event horizon. Thus, proceeding as above, the entropyon the event horizon can be written as [28] S E = A E G − π Z R A R E − ǫ ! (cid:18) HR E + 1 HR E − (cid:19) ( ρ e + p e ) dR E (31)which again shows that the leading term for entropy is the usual Bekenstein entropy. III. BASIC EQUATIONS IN MASSIVE GRAVITY
In recent times, a new modified theory of gravity has been constructed by adding a small mass tothe graviton. In classical field theory, there arises a basic question of whether a consistent extensionof general relativity (GR) by a mass term is possible or not. Fierz and Pauli (FP) initiated thisattempt to construct a theory of gravity with massive graviton [29]. They added a quadratic massterm m ( h µν h µν − h ) to the action (for linear gravitational perturbations) and as a result there wasa violation of gauge invariance in GR. Further, the linear theory with FP-mass does not match withGR in the zero mass limit and this leads to the contradiction with solar system tests due to the vDVZdiscontinuity [30, 31]. However, introduction of nonlinear interactions using Vanishtein mechanism [32]might overcome this problem. This idea was subsequently used by Dvali, Gabadadze and Porrati [33]to construct a higher dimensional model of massive gravity which admits a self-accelerating solutionwith dust matter and the theory modifies GR at the cosmological scale.On the other way, the Bouldware-Deser (BD) ghost [34] in the formulation with St¨ u ckelberg fieldcan be eliminated by adding nonlinear interactions with higher derivatives, order by order in pertur-bation theory. Recently, de Rham, Gabadadze and Tolley (dGRT) formulated a nonlinear massivegravity theory which is free from BD ghost [35]. Subsequently, various cosmological solutions [36–43]have been evaluated using dGRT-massive gravity theory. To ensure that no ghost appears in thedecoupling limit, the total action takes the form [35] S = 116 πG Z d x √− g [ R + m g U ] + S m , (32)where m g stands for graviton mass and the nonlinear higher derivative term U corresponding tomassive graviton has the expression [19,27] U = U + α U + α U (33)with U = [ κ ] − [ κ ] , (34) U = [ κ ] − κ ][ κ ] + 2[ κ ] , (35) U = [ κ ] − κ ] [ κ ] + 8[ κ ][ κ ] − κ ] . (36)The tensor κ µν is defined as κ µν = δ µν − q ∂ µ φ α ∂ ν φ b f ab , (37)where φ α stands for Stu¨ckelberg field and the reference metric f ab is usually taken to be Minkowskianmetric, i.e., f ab = diag ( − , , , κ ] = tr κ µν , [ κ ] = ( tr κ µν ) , [ κ ] = tr κ µν κ νλ (38)and so on.For convenience, if the unitary gauge φ a ( x ) = x µ δ µa is chosen, then the metric tensor stands for theobservable describing the 5 degrees of freedom of the massive graviton. However, for the Minkowskianreference metric, the theory does not even admits any nontrivial flat homogeneous and isotropic(FLRW) solution [45], so in the present work, we choose the reference metric f ab as the de Sittermetric. As a result, it is possible to have the flat, open or closed cosmologies by suitable slicing of deSitter model [46] and also this choice of reference metric eliminates the problem of ’no-go’ theorem[36]. Hence the reference metric is taken as f ab dφ a dφ b = − dT + b k ( T ) γ ij ( X ) dX i dX j (39)with b ( T ) = e H c T , b − ( T ) = H − c sinh ( H c T ) , b t ( T ) = H − c cosh ( H c T ) . (40)Note that in the limit H c →
0, the Minkowski metric is recovered for flat and open cases: b = 1, b − = T , while the latter case corresponds to the Milne metric for flat geometry.Now, choosing the Stu¨ckelberg field as φ = T = f ( T ) , φ i = X i = x i , (41)one sees that the cosmological symmetries are satisfied and the symmetric (0,2) tensorΣ µν = f ab ∂ µ φ a ∂ ν φ b (42)becomes a homogeneous and isotropic tensor of the formΣ µν = diag ( − ˙ f , b k f ( t ) γ ij ) . (43)Thus the elements of the κ matrix has the simple form κ = 1 − ζ f ˙ fN , κ ji = (cid:18) − b k ( f ) a (cid:19) δ i j , κ i = 0 , κ i = 0 , (44)where ζ f denotes the sign of ’ f ’. Hence the explicit expression for the nonlinear higher derivative term(representing the potential for graviton) in the Lagrangian is L mg = √− gU = ( a − b k ( f )) h N n a (4 α + α + 6) − a (5 α + 2 α + 3) b k ( f ) + ( α + α ) b k ( f ) oi − ( a − b k ( f )) h ζ f ˙ f n a (3 α + α + 3) − a (3 α + 2 α ) b k ( f ) + α b k ( f ) + α b k ( f ) oi . (45)Considering variation of ’ f ’ in the Lagrangian, the equation of motion for f ( t ) can be written as h (3 α + α + 3) a − α + α + 1) ab k ( f ) + ( α + α ) b k ( f ) i (cid:18) ˙ aN − ζ f b ′ k ( f ) (cid:19) = 0 . (46)Hence, solving for b k ( f ), one gets b k ( f ( t )) = µ ± a ( t ) , (47)where µ ± = 1 + 2 α + α ± q α + α − α α + α . (48)Thus it is possible to have f ( t ) from Eq. (47), provided b k is invertible. Note that in the Minkowskiandomain (i.e., f ab = η ab ), we have b ( f ) = 1 for flat case and hence no solution for ’ f ’ is possible whilethere are two branches of solutions for open case ( b − ( f ) = f ) [36, 38]. Further, from the remainingpart of the evolution equation (46), one gets [41, 47] ζ f b ′ k ( f ) = ˙ aN . (49)So, similarly as before, non-trivial solutions are possible only if inversion of b ′ k ( f ) is possible. Notethat this case has no analogue with Minkowskian reference metric because there does not exist any0solution for both flat and open cases. However, choosing b k ’s from Eq. (40), one gets an explicitsolution for f ( t ) as f ( t ) = H − c ln (cid:18) H ( t ) a ( t ) H c (cid:19) , (50)where H = N ˙ aa is the usual Hubble parameter and N is the lapse function (which sets to unity in thesubsequent steps).Thus for the present FLRW metric, the usual Einstein-Hilbert (EH) component has the explicitform L EH = − a aN + 3 κN a (51)and in addition to the massive gravity part, there is Lagrangian L m corresponding to ordinary cos-mological matter. So the variation of the total Lagrangian with respect to the lapse function N andthe scale factor a gives the first and second Friedmann equations as3 (cid:18) H + κa (cid:19) = 8 πG ( ρ + ρ e ) (52)and ˙ H − κa = − πG ( ρ + p m + ρ e + p e ) , (53)where (as in the previous section) ( ρ , p ) are the energy density and thermodynamic pressure ofthe cosmic fluid and ( ρ e , p e ) are the effective energy density and thermodynamic pressure due tocontribution from the massive gravity part with explicit expressions ρ e = m g πGa ( b k ( f ) − a ) n (4 α + α + 6) a − a (5 α + 2 α + 3) b k ( f ) + ( α + α ) b k ( f ) o p e = m g πGa hn α + α + 6 − (3 α + α + 3) ˙ f o a − n α + α + 3 − (2 α + α + 1) ˙ f o ab k ( f ) i + m g πGa hn α + α − ( α + α ) ˙ f o b k ( f ) i . (54)Now, based on evidences from the WMAP data [48], in the present work, we shall restrict ourselvesto flat FLRW model of the Universe. So using solution (50) for f ( t ), the simplified form of ρ e and p e are the following: ρ e = m g πG " − α + 9 β HH c − γ H H c + 3 δ H H c p e = − ρ e + m g πG ˙ HH HH c " − β + 2 γ HH c − δ H H c , (55)where α = 1 + 2 α + 2 α , β = 1 + 3 α + 4 α , γ = 1 + 6 α + 12 α , δ = α + 4 α . (56)1Thus the present massive gravity theory contains three free parameters namely m g , α and α andit should be noted that there is no longer any contribution of massive graviton to the energy density(i.e., ρ e = 0) if H = H c . IV. A THERMODYNAMICAL STUDY IN MASSIVE GRAVITY THEORY
This section deals with universal thermodynamics of massive gravity theory in the background offlat FLRW model. So the modified Friedmann equations take the form H = 8 πG ρ + ρ e )˙ H = − πG ( ρ + p + ρ e + p e ) . (57)Now the expression for entropy on the apparent and the event horizon (after some algebra) canbe obtained in massive gravity theory (for details of calculation, one may refer to Appendix A) fromEqs. (29) and (31) as S A = A A G − πm g GH c (cid:20) βR A − γH c R A + 3 δH c R A (cid:21) (58)and S E = A E G − πm g GH c Z (cid:20) R E ( HR E + 1)2 − ˙ R A (cid:26) − βH + 2 γH c H − δH c H (cid:27)(cid:21) dH (59)respectively. One should note that the first term in both the above expressions is the usual Bekensteinentropy for the corresponding horizon, A A and A E being the area of the horizons.The expression for the entropy variation of the fluid bounded by any horizon (given in Eq. (3))can be expressed in a general form as (for details of calculation, one may refer to Appendix B)˙ S fh = 4 πR h T f ( ρ + p )( ˙ R h − HR h ) . (60)2For a flat model of the Universe, the radii of the two horizons (namely apparent and event) are relatedby the inequality R A = H < R E . THe thermodynamical parameters namely (Bekenstein) entropyand (Hawking) temperature on the apparent horizon are given by S A = πR A G and T A = 12 πR A . (61)However, following Ref. [14], the entropy and the temperature on the event horizon are written as S E = πR E G and T E = R E πR A = H R E π . (62)Thus, taking derivatives of Eqs. (58) and (59), adding them respectively to Eq. (60) and substitut-ing the expressions of T A and T E from Eqs. (61) and (62), one can evaluate the total entropy variationfor Universe bounded by the apparent and the event horizons as (considering the temperature of thehorizons and that of the fluid inside them to be equivalent)˙ S T A = 3 π G (1 + ω ) " (cid:26) πG Ω m (cid:18) ω − ω (cid:19)(cid:27) R A − m g H c (cid:18) βR A − γH c R A + 3 δH c (cid:19) (63)and ˙ S T E = πG h R E ( HR E − − m g H c (cid:18) ω − ω (cid:19) R E ( HR E + 1) (cid:26) βR A − γH c + 3 δH c R A (cid:27) − πG Ω m (cid:18) ω − ω (cid:19) R E i (64)respectively. We have assumed that the Universe is filled with a perfect fluid having constant equationof state, i.e., p = ωρ where ω is a constant and Ω m = ρ H is the usual density parameter. Thevelocities of the apparent and the event horizons are given by v A = ˙ R A = − ˙ HH = 32 (1 + ω ) (65)and v E = ˙ R E = ( HR E − . (66)Differentiating Eqs. (63) and (64) again, we obtain¨ S T A = 9 π G (1 + ω ) " πG Ω m (cid:18) ω − ω (cid:19) − m g H c (cid:18) βR A − γH c (cid:19) (67)and ¨ S T E = πG h n (1 − ω ) H R E − HR E + 2 o − m g H c (cid:18) ω − ω (cid:19) n (cid:18) βR A − γH c + 3 δH c R A (cid:19) × (cid:16) (1 − ω ) H R E − HR E − (cid:17) + 9(1 + ω ) β − δH c R A ! R E ( HR E + 1) o − πG Ω m (cid:18) ω − ω (cid:19) ( HR E − i . (68)3As the expressions for the first and second order time variation of the total entropy are verycomplicated, so it is not possible to predict the validity of GSLT and thermodynamical equilibrium(given by inequalities (1)) analytically. Hence we have examined their validity from the graphicalrepresentation in Figs. 1-4 for various choices of the parameters involved. V. BRIEF DISCUSSION AND FINAL COMMENTS
This paper deals with the study of Universal thermodynamics, particularly the generalized secondlaw of thermodynamics (GSLT) and the thermodynamical equilibrium (TE) for Universe bounded byan apparent/event horizon in massive gravity under the assumption that the first law (i.e., Clausiusrelation) holds. Cosmological solutions of massive gravity theory as a viable alternative to describe thepresent day cosmological observations have been widely investigated. But unfortunately, it was found[45] that all homogeneous and isotropic solutions in dGRT theory are unstable. Moreover, there doesnot exist any nontrivial FLRW solution with Minkowskian reference metric. So in the present work, wehave chosen the reference metric to be de Sitter for which the theory does admit flat FLRW solutions(see Eq. (50)) but it is unstable in nature. Thus the aim of this work was to analyze this cosmologicalsystem in the thermodynamical perspective. We have established modified forms for entropy (Eqs.(61) and (62)) associated with the apparent/event horizon using methods proposed by Hayward andCai. Using these equations and the Gibb’s equation, we have been able to evaluate the total entropyvariation and its derivative for Universe bounded by the apparent/event horizon and filled with perfectfluid having constant equation of state parameter ω . In Figs. 1 and 2, variation of the total entropyhas been plotted against ω and ( H c = 1 , . , .
3) for Universe bounded by the apparent and the eventhorizon respectively over one Hubble scale and at R E = 2. The other parameters have been chosen as4Ω m = 0 . m g = 1, α = and α = . We see that GSLT holds for the apparent horizon when thefluid is not beyond ΛCDM while for the event horizon, GSLT holds when the fluid is almost exoticin nature (i.e., ω < − ). On the other hand, Figs. 3 and 4 depict the TE for Universe boundedby the apparent and the event horizon respectively for similar values of the parameters as in Figs. 1and 2. It is evident from Fig. 3 that TE holds unconditionally for the apparent horizon except at ω = −
1, where we have the limiting situation. Fig. 4 shows that validity of TE for the event horizonis restricted only for normal fluid. Therefore, based on the above analysis, we can conclude that inthe present unstable cosmological scenario (in massive gravity theory), the apparent horizon is morefavourable as compared to the event horizon from thermodynamical perspective in massive gravity,which is in contrast to our earlier observations [17].
ACKNOWLEDGMENTS
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APPENDIX A
Using Eq. (55) in Eq. (29), one obtains S A = A A G − πG Z m g ˙ HH HH c " − β + 2 γ HH c − δ H H c HR A dt = A A G − πm g GH c Z " − β + 2 γ HH c − δ H H c dHH = A A G − πm g GH c (cid:20) βR A − γH c R A + 3 δH c R A (cid:21) , which is Eq. (58).Using Eq. (55) in Eq. (31), the entropy of the event horizon takes the form S E = A E G − π G Z R A R E − ǫ ! (cid:18) HR E + 1 HR E − (cid:19) " m g ˙ HH HH c − β + 2 γ HH c − δ H H c ! dR E = A E G − πm g GH c Z R E (cid:18) HR E + 11 − ǫ (cid:19) " − β + 2 γ HH c − δ H H c dHH = A E G − πm g GH c Z (cid:20) R E ( HR E + 1)1 − ˙ R A (cid:26) − βH + 2 γH H c − δHH c (cid:27)(cid:21) dH, which is Eq. (59). APPENDIX B
From Eq. (3), T f dS fh = dE f + pdV h or, T f dS fh dt = ˙ ρ m V h + ( ρ m + p m ) dV h dtor, T f ˙ S fh = − H ( ρ m + p m ) · πR h + 4 π ( ρ m + p m ) R h ˙ R h or, ˙ S fh = 4 πR h T f ( ρ m + p m )( ˙ R h − HR h ) ,,