Eur. Phys. J. C (2010) 69: 265–269DOI 10.1140/epjc/s10052-010-1372-9

Regular Article - Theoretical Physics

Thermodynamics of apparent horizon and modified Friedmannequations

Ahmad Sheykhia

Department of Physics, Shahid Bahonar University, P.O. Box 76175, Kerman, IranResearch Institute for Astronomy and Astrophysics of Maragha (RIAAM), Maragha, Iran

Received: 5 April 2010 / Revised: 8 May 2010 / Published online: 25 June 2010© Springer-Verlag / Società Italiana di Fisica 2010

Abstract Starting from the first law of thermodynamics,dE = Th dSh + W dV , at the apparent horizon of a FRWuniverse, and assuming that the associated entropy with ap-parent horizon has a quantum-corrected relation, S = A

4G−

α ln A4G

+β 4GA

, we derive modified Friedmann equations de-scribing the dynamics of the universe with any spatial curva-ture. We also examine the time evolution of the total entropyincluding the quantum-corrected entropy associated with theapparent horizon together with the matter field entropy in-side the apparent horizon. Our study shows that, with thelocal equilibrium assumption, the generalized second law ofthermodynamics is fulfilled in a region enclosed by the ap-parent horizon.

1 Introduction

The pioneer study on the deep connection between grav-ity and thermodynamics was done by Jacobson [1], whoshowed that the gravitational Einstein equation can be de-rived from the relation between the horizon area and entropy,together with the Clausius relation δQ = T δS. Further stud-ies on the connection between gravity and thermodynamicshas been investigated in various gravity theories [2–10]. Inthe cosmological context, attempts to disclose the connec-tion between Einstein gravity and thermodynamics were car-ried out in [11–22]. It was shown that the differential formof the Friedmann equation in the Friedmann–Robertson–Walker (FRW) universe can be written in the form of thefirst law of thermodynamics on the apparent horizon. Theprofound connection provides a thermodynamical interpre-tation of gravity which makes it interesting to explore thecosmological properties through thermodynamics. Investi-gations on the deep connection between gravity and thermo-

a e-mail: sheykhi@mail.uk.ac.ir

dynamics have recently been extended to braneworld sce-narios [23–26].

It is interesting to note that Friedmann equations, in Ein-stein’s gravity, can be derived by applying the Clausius re-lation to the apparent horizon of FRW universe, in whichentropy is assumed to be proportional to its horizon area,S = A/4G [13]. However, this definition for entropy can bemodified from the inclusion of quantum effects, motivatedfrom the loop quantum gravity (LQG). The quantum correc-tions provided to the entropy-area relationship leads to thecurvature correction in the Einstein–Hilbert action and viceversa [27, 28]. The corrected entropy takes the form [29–32]

Sh = A

4G− α ln

A

4G+ β

4G

A, (1)

where α and β are positive dimensionless constants of orderunity. The exact values of these constants are not yet deter-mined and still an open issue in loop quantum cosmology.These corrections arise in the black hole entropy in LQGdue to thermal equilibrium fluctuations and quantum fluctu-ations [33, 34]. It is important to note that in the literaturedifferent kind of modification of entropy expression havestudied in classical level for various modified gravity the-ories [35–38]. The log correction to the area–entropy rela-tion appears to have an almost universal status, having beenderived from multiple different approaches to the calcula-tion of entropy from counting microscopic states in differentquantum gravity models. Let us stress here that although inthe literature there is doubt about the second correction termin entropy-corrected relation, however, it is widely believed[29–32] that the next quantum correction term to black holeentropy have the form 4G/A, which leads to the reasonablecorrection terms to Newton’s law of gravitation [39] and willalso lead to the corrected modified Friedmann equation aswe will show in this paper.

Besides, if thermodynamical interpretation of gravitynear apparent horizon is generic feature, one needs to ver-

266 Eur. Phys. J. C (2010) 69: 265–269

ify whether the results may hold not only for more generalspacetimes but also for the other principles of thermodynam-ics, especially for the generalized second law of thermody-namics. The generalized second law of thermodynamics is auniversal principle governing the universe. The generalizedsecond law of thermodynamics in the accelerating universeenveloped by the apparent horizon has been studied exten-sively in [40–44]. For other gravity theories, the generalizedsecond law has also been considered in [45, 46].

The aim of this paper is twofold. The first is to derivemodified Friedmann equations by applying the first law ofthermodynamics, dE = Th dSh+W dV , at apparent horizonof a FRW universe and assuming the apparent horizon has anentropy expression like (1). The other is to see whether thequantum-corrected-entropy–area relation together with thematter field entropy inside the apparent horizon will satisfythe generalized second law of thermodynamics.

2 Modified Friedmann equation from the first law ofthermodynamics

We consider a homogeneous and isotropic FRW universewhich is described by the line element

ds2 = hμν dxμ dxν + r2(dθ2 + sin2 θ dφ2), (2)

where r = a(t)r , x0 = t , x1 = r , the two dimensional metrichμν=diag (−1, a2/(1 − kr2)). Here k denotes the curvatureof space with k = 1,0,−1 corresponding to open, flat, andclosed universes, respectively. The dynamical apparent hori-zon, a marginally trapped surface with vanishing expansion,is determined by the relation hμν∂μr∂ν r = 0. Straightfor-ward calculation gives the apparent horizon radius for theFRW universe

rA = 1√

H 2 + k/a2. (3)

The associated temperature with the apparent horizon canbe defined as T = κ/2π , where κ is the surface gravity κ =

1√−h∂μ(

√−hhμν∂μν r). Then one can easily show that thesurface gravity at the apparent horizon of FRW universe canbe written as

κ = − 1

rA

(1 −

˙rA

2HrA

). (4)

When ˙rA ≤ 2HrA, the surface gravity κ ≤ 0, which leadsthe temperature T ≤ 0 if one defines the temperature of theapparent horizon as T = κ/2π. Physically it is not easy toaccept the negative temperature, the temperature on the ap-parent horizon should be defined as T = |κ|/2π . Recently,the connection between temperature on the apparent horizonand the Hawking radiation has been considered in [47, 48],

which gives a more solid physical implication of the temper-ature associated with the apparent horizon.

Suppose the matter source in the FRW universe is a per-fect fluid with stress-energy tensor

Tμν = (ρ + p)uμuν + pgμν, (5)

where ρ and p are the energy density and pressure, respec-tively. The energy conservation law then leads to

ρ + 3H(ρ + p) = 0, (6)

where H = a/a is the Hubble parameter. Following [49, 50],we define the work density as

W = −1

2T μνhμν. (7)

In our case it becomes

W = 1

2(ρ − p). (8)

The work density term is regarded as the work done by thechange of the apparent horizon. Assuming the first law ofthermodynamics on the apparent horizon is satisfied and hasthe form

dE = Th dSh + W dV, (9)

where Sh is the quantum-corrected entropy associated withthe apparent horizon which has the form (1). One can seethat in (9), the work density is replaced with the negativepressure if we compare with the standard first law of ther-modynamics, dE = T dS−pdV . For a pure de Sitter space,ρ = −p, then the work term reduces to the standard −p dV

and we obtain exactly the standard first law of thermody-namics.

We also assume E = ρV is the total energy content of theuniverse inside a 3-sphere of radius rA, where V = 4π

3 r3A

is the volume enveloped by 3-dimensional sphere with thearea of apparent horizon A = 4πr2

A. Taking differential formof the relation E = ρ 4π

3 r3A for the total matter and energy

inside the apparent horizon, we get

dE = 4πr2Aρ drA + 4π

3r3Aρ dt. (10)

Using the continuity equation (6), we obtain

dE = 4πr2Aρ drA − 4πHr3

A(ρ + p)dt. (11)

Taking the differential form of the corrected entropy (1), wehave

dSh = 2πrA

G

[1 − αG

πr2A

− βG2

π2r4A

]drA. (12)

Eur. Phys. J. C (2010) 69: 265–269 267

Inserting (8), (11) and (12) in the first law (9) and using therelation between temperature and surface gravity, we can getthe differential form of the modified Friedmann equation

1

4πG

drA

r3A

[1 − αG

πr2A

− βG2

π2r4A

]= H(ρ + p)dt. (13)

Using the continuity equation (6), we can rewrite it as

−2drA

r3A

[1 − αG

πr2A

− βG2

π2r4A

]= 8πG

3dρ. (14)

Integrating (14) yields

1

r2A

− αG

2πr4A

− βG2

3π2r6A

= 8πG

3ρ, (15)

where an integration constant, which is just the cosmologi-cal constant, has been absorbed into the energy density ρ.Substituting rA from (3) we obtain the entropy-correctedFriedmann equation

H 2 + k

a2− αG

2π

(H 2 + k

a2

)2

− βG2

3π2

(H 2 + k

a2

)3

= 8πG

3ρ. (16)

In this way we derive the entropy-corrected Friedmannequation by starting from the first law of thermodynamics,dE = Th dSh + W dV , at apparent horizon of a FRW uni-verse, and assuming that the associated entropy with ap-parent horizon has a quantum-corrected relation (1). In theabsence of the correction terms (α = 0 = β), one recoversthe well-known Friedmann equation in standard cosmology.Since the last two terms in (16) can be comparable to the firstterm only when a is very small, the corrections make senseonly at early stage of the universe where a → 0. When theuniverse becomes large, the entropy-corrected Friedmannequation reduces to the standard Friedmann equation.

It is important to note that in the literature many differentmodifications of entropy and therefore of Friedmann equa-tions are studied in classical modified gravity theories. Forexample, in [51] the modified gravity with lnR or R−n termswhich grow at small curvature was discussed. It was shown[51] that such a model may eliminate the need for dark en-ergy and may provide the current cosmic acceleration. It wasalso demonstrated that R2 terms are important not only forearly time inflation but also to avoid the instabilities andthe linear growth of the gravitational force. Thus, modifiedgravity with R2 term seems to be viable classical theory. Itwas also argued in [38, 52] that the modified gravity wheresome arbitrary function of Gauss–Bonnet term is added toEinstein action can explain the dark energy dominated uni-verse. It was shown that such theory may pass solar system

tests and can describe the most interesting features of late-time cosmology such as the transition from deceleration toacceleration, crossing the phantom divide, current accelera-tion with effective (cosmological constant, quintessence orphantom) equation of state of the universe. In [53] the mod-ification of the Friedmann equations which may be causedby f (R) gravity, string-inspired scalar-Gauss–Bonnet, mod-ified Gauss–Bonnet theories, and ideal fluid with the inho-mogeneous equation of state. It was demonstrated [53] thatthe history of the expansion of the universe can be recon-structed through such a universal formulation. Further inves-tigations on the cosmological implications of the modifiedtheory of gravity have been carried out in [54–58].

It is also worth mentioning that (16) is in complete agree-ment with the result of [59]. However, our derivation is quitedifferent from [59]. Let us stress the difference between hereand [59]. First of all, the authors of [59] have derived mod-ified Friedmann equations by applying the first law of ther-modynamics, T dS = −dE, to the apparent horizon of aFRW universe with the assumption that the apparent horizonhas temperature T = 1/2πrA and corrected entropy like (1).It is worthy to note that the notation dE in [59] is quite dif-ferent from the same we used in the present work. In [59],−dE is actually just the heat flux δQ in [1] crossing theapparent horizon within an infinitesimal internal of time dt .But, here dE is change in the matter energy inside the ap-parent horizon. Besides, in [59] the apparent horizon radiusrA has been assumed to be fixed. Thus, the temperature ofapparent horizon can be approximated to T = 1/2πrA andthere is no the term of volume change in it. But, here, wehave used the matter energy E inside the apparent horizonand the apparent horizon radius changes with time. This isthe reason why we have included the term WdV in the firstlaw (9). Indeed, the term 4πr2

Aρ drA in (11) contributes tothe work term, while this term is absent in dE of [59]. Thisis consistent with the fact that in thermodynamics the workis done when the volume of the system is changed. We haveassumed that drA is the infinitesimal change in the radius ofthe apparent horizon in a small time interval dt which causesa small change dV of volume inside the apparent horizon.Since the matter energy E is directly related to the radius ofthe apparent horizon, therefore, the change of apparent hori-zon radius will change the energy dE inside the apparenthorizon.

3 Generalized second law of thermodynamics

In this section we turn to investigate the validity of the gen-eralized second law of thermodynamics in a region enclosed

268 Eur. Phys. J. C (2010) 69: 265–269

by the apparent horizon. Differentiating (15) with respect tothe cosmic time and using (6) we get

−2 ˙rAr3A

[1 − αG

πr2A

− βG2

π2r4A

]= −8πGH(ρ + p). (17)

Solving for ˙rA we find

˙rA = 4πGHr3A(ρ + p)

[1 − αG

πr2A

− βG2

π2r4A

]−1

. (18)

One can see from the above equation that ˙rA > 0 providedthe dominant energy condition, ρ +p > 0, holds. Let us nowturn to find out ThSh:

ThSh = 1

2πrA

(1 −

˙rA

2HrA

)d

dt

(A

4G− α ln

A

4G+ β

4G

A

).

(19)

After some simplification and using (18) we obtain

ThSh = 4πHr3A(ρ + p)

(1 −

˙rA

2HrA

). (20)

In the accelerating universe the dominant energy conditionmay violate, ρ + p < 0, indicating that the second law ofthermodynamics ,Sh ≥ 0, does not hold. However, as we willsee below the generalized second law of thermodynamics,Sh + Sm ≥ 0, is still fulfilled throughout the history of theuniverse. From the Gibbs equation we have [60]

Tm dSm = d(ρV ) + p dV = V dρ + (ρ + p)dV, (21)

where Tm and Sm are, respectively, the temperature and theentropy of the matter fields inside the apparent horizon. Welimit ourselves to the assumption that the thermal systembounded by the apparent horizon remains in equilibrium sothat the temperature of the system must be uniform and thesame as the temperature of its boundary. This requires thatthe temperature Tm of the energy inside the apparent horizonshould be in equilibrium with the temperature Th associatedwith the apparent horizon, so we have Tm = Th [60]. Thisexpression holds in the local equilibrium hypothesis. If thetemperature of the fluid differs much from that of the hori-zon, there will be spontaneous heat flow between the hori-zon and the fluid and the local equilibrium hypothesis willno longer hold. Therefore from the Gibbs equation (21) wecan obtain

ThSm = 4πr2A

˙rA(ρ + p) − 4πr3AH(ρ + p). (22)

To check the generalized second law of thermodynamics, wehave to examine the evolution of the total entropy Sh + Sm.Adding (20) and (22), we get

Th(Sh + Sm) = 2πr2A(ρ + p) ˙rA = A

2(ρ + p) ˙rA, (23)

where A is the apparent horizon area. Substituting ˙rA from(18) into (23) we find

Th(Sh + Sm)

= 2πGAHr3A(ρ + p)2

[1 − αG

πr2A

− βG2

π2r4A

]−1

. (24)

It is important to note that the expression in the bracket of(24) is positive at the present time, i.e.,

[1 − αG

πr2A

− βG2

π2r4A

]> 0. (25)

This is due to the fact that at the present time rA � 1 whileα ∼ O(1), β ∼ O(1) and G ∼ 10−11, thus αG

πr2A

� 1 and

βG2

π2 r4A

� 1. At the early time where rA → 0 the generalized

second law of thermodynamics may be violated but in thatcase the local equilibrium hypothesis is failed too. Besides,from the physical point of view, the effect of the correctionterms on the entropy should be less than uncorrected term.Thus, the second and third terms on the right hand side of (1)and (12) should be much smaller than the first term, other-wise these terms cannot be regarded as the correction terms.For all above reasons we can expand the right hand side of(24), up to the linear order of α and β ,

Th(Sh + Sm)

= 2πGAHr3A(ρ + p)2

[1 + αG

πr2A

+ βG2

π2r4A

]. (26)

The right hand side of the above equation cannot be neg-ative throughout the history of the universe, which meansthat Sh + Sm ≥ 0 always holds. This indicates that for a uni-verse with any spacial curvature the generalized second lawof thermodynamics is fulfilled in a region enclosed by theapparent horizon.

4 Conclusions

In summary, applying the first law of thermodynamics,dE = Th dSh + W dV , to apparent horizon of a FRW uni-verse with any spatial curvature and assuming that the

apparent horizon has temperature T = 12πrA

(1 − ˙rA

2HrA),

and a quantum-corrected-entropy–area relation, Sh = A4G

−α ln A

4G+ β 4G

A, we are able to derive modified Friedmann

equations governing the dynamical evolution of the uni-verse. We have also investigated the validity of the gen-eralized second law of thermodynamics for the FRW uni-verse with any spatial curvature. We have shown that, whena thermal system bounded by the apparent horizon remains

Eur. Phys. J. C (2010) 69: 265–269 269

in equilibrium with its boundary such that Tm = Th, the gen-eralized second law of thermodynamics is fulfilled in a re-gion enclosed by the apparent horizon. The validity of thegeneralized second law of thermodynamics for quantum-corrected-entropy–area relation further supports the thermo-dynamical interpretation of gravity and provides more confi-dence on the profound connection between gravity and ther-modynamics.

It is worth noting that although we derived modifiedFriedmann equations corresponding to the corrected-entropy–area relation (1) by applying the first law of ther-modynamics to apparent horizon, it would be of great inter-est to see whether one is able to get modified Einstein fieldequation by following Jacobson argument [1]. This study isof great interest and further shows that given a thermody-namical relation between entropy and geometry, one is ableto derive corresponding modified Einstein field equation,showing an interesting connection between them.

Finally, we would like to mention that the higher orderterms of (H 2 + k/a2) in the modified Friedmann equations(16) only becomes important at early time of the universe.They may influence the number of e-folds of inflation, orthey may give corrected upper bound on the number of e-folds following the holographic principle. These should beexamined carefully. Equation (16) does not look to influencethe late-time cosmology. The detail of this study will be ad-dressed elsewhere.

Acknowledgements I thank the anonymous referees for construc-tive comments. I am also grateful to Prof. B. Wang for helpful discus-sions. This work has been supported by Research Institute for Astron-omy and Astrophysics of Maragha, Iran.

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