Research Article Thermodynamics in Modified Gravity with ... · /2C 9 2 )). e Bekenstein-Hawking...

Hindawi Publishing CorporationAdvances in High Energy PhysicsVolume 2013, Article ID 947898, 9 pageshttp://dx.doi.org/10.1155/2013/947898

Research ArticleThermodynamics in Modified Gravity withCurvature Matter Coupling

M. Sharif and M. Zubair

Department of Mathematics, University of the Punjab, Quaid-e-Azam Campus, Lahore 54590, Pakistan

Correspondence should be addressed to M. Sharif; [email protected]

Received 15 August 2013; Revised 27 September 2013; Accepted 1 October 2013

Academic Editor: Kingman Cheung

Copyright © 2013 M. Sharif and M. Zubair. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

The first and generalized second laws of thermodynamics are studied in 𝑓(𝑅, 𝐿𝑚) gravity, a more general modified theory with

curvature matter coupling. It is found that one can translate the Friedmann equations to the form of first law accompanied withentropy production term.This behavior is due to the nonequilibrium thermodynamics in this theory. We establish the generalizedsecond law of thermodynamics and develop the constraints on coupling parameters for two specific models. It is concluded thatlaws of thermodynamics in this modified theory are more general and can reproduce the corresponding results in Einstein, 𝑓(𝑅)gravity, and 𝑓(𝑅) gravity with arbitrary as well as nonminimal curvature matter coupling.

1. Introduction

The rapid growth of observational measurements on expan-sion history reveals the expanding paradigm of the universe.This fact is based on accumulative observational evidencesmainly from type Ia supernova and other renowned sources[1–3]. The expanding phase implicates the presence of repul-sive force which compensates the attractiveness property ofgravity on cosmological scales. This phenomenon may betranslated as the existence of exotic matter components andmost acceptable understanding for such enigma is termedas dark energy (DE) having large negative pressure. Severalapproaches have been proposed to explain transition frommatter dominated to accelerated expansion era. The mostcommon representative for DE is the cosmological constanthaving equation of state (EoS) 𝜔 = −1 [4]. However, thereare some serious issues to explain its mystery in any theory.Furthermore, observations show that EoS may cross thephantomdivide andWMAP5 data set the bound for the valueof 𝜔 in the range −1.11 < 𝜔 < −0.86 [5]. In Einstein gravity,the issue of accelerated expansion is explained in the spiritof various DE models such as Chaplygin gas, quintessence,phantom, quintom, and Holographic DE models [6–10].

The other promising way to deal with the cosmic expan-sion is the modification of Einstein-Hilbert action where it is

assumed that Einstein gravity breaks down on large scales.The 𝑓(𝑅) theory is one of the theoretical models in thiscontext which has attained significant attention to explain thecosmic acceleration [11, 12]. In this modification, the simplestchoice is to replace scalar curvature by𝑅𝑛, for 𝑛 > 1, it exhibitsthe de Sitter behavior for early times while for 𝑛 = −1, 3/2, itexplains the accelerated expansion [13, 14]. The 𝑓(𝑅) theoryinvolving inverse power of scalar curvature can be a handycandidate of DE for which 𝑓(𝑅) = (𝑅−(𝜇/𝑅𝑛)) (𝜇 > 0, 𝑛 > 0)[15, 16]. However, this model has been ruled out due to localgravity constraints and matter instability condition. At thesame time, 𝑓(𝑅) models can satisfy solar system constraints(for more general viable models see [17–19]).

In most modified gravitational theories, the Einsteingravity is generalized by changing the geometric part whereasmatter part receives no attention. Bertolami et al. [20]presented the generalization of 𝑓(𝑅) theory by introducingnon-minimal curvature matter coupling and it is extended toLagrangian involving arbitrary function ofmatter Lagrangiandensity [21]. Curvature matter coupling results in non-geodesic motion of test particles and hence an extra forceorthogonal to four-velocity originates [20]. The correspon-dence between non-minimal coupling in this theory andscalar-tensor theory has been developed in [22], and it wasshown that non-minimally coupled theory would imply two

2 Advances in High Energy Physics

scalar fields. Nesseris [23] studied matter density perturba-tions to constrain this theory from growth factor as well asweak lensing observations. This class of models in modifiedtheory has been extensively studied in the literature [24–27](for review on modified theories with curvature matter seealso [28]).

Curvature matter coupling has also been considered byintroducing a Lagrangian having arbitrary dependence onthe trace of energy-momentum tensor named as 𝑓(𝑅, 𝑇)[29]. In this theory, motion of test particles is nongeodesicproducing extra force and cosmic acceleration may resultdue to mutual contribution both from geometric and materparts. Recently, this theory has been under consideration andsome interesting results have been obtained [30–36]. Thismodel is further generalized by incorporating the possiblecoupling between energy-momentum as well as the Riccitensor and Lagrangian of the form 𝑓(𝑅, 𝑇, 𝑅

𝜇]𝑇𝜇]) which is

established [37, 38]. In a recent paper [39], we have presentedthe field equations for a more general case and formulatedenergy conditions corresponding to this modified theory.The specific functional forms of Lagrangian 𝑓(𝑅, 𝑇, 𝑅

𝜇]𝑇𝜇])

have been constrained using the energy conditions and theDolgov-Kawasaki instability.

The thermodynamic behavior of accelerating universedriven by DE is one of the major concerns in cosmology.It has been shown that Einstein field equations can beobtained using the relation 𝛿𝑄 = 𝑇𝑑𝑆 by considering theproportionality of horizon area and entropy [40–43]. Therelation between thermodynamics and gravity has beentested in Einstein as well as Gauss-Bonnet and Lovelockgravities [44]. Cai and Cao [45] showed that Friedmannequations in braneworld scenario can be cast to the form offirst law of thermodynamics at the apparent horizon. Thiswork is also extended in the framework of warped DGPbraneworld [46] and Gauss-Bonnet Braneworld [47].

Akbar and Cai [48] discussed the first law of thermo-dynamics in scalar tensor and 𝑓(𝑅) gravities at the appar-ent horizon of FRW universe. They proposed that equilib-rium thermodynamics can be achieved in these theories byincorporating the curvature contribution term to effectiveenergymomentum tensor. However, in nonlinear theoriesof gravity there is an issue of non-equilibrium picture ofthermodynamics which was first suggested by Eling et al.[49] and has been discussed under different circumstances[50, 51]. Cai and Cao [52] found that in scalar tensor theoriesthermodynamics associated with the apparent horizon of theFRW universe results in non-equilibrium description whichmodifies the standardClausius relation.The non-equilibriumtreatment of thermodynamics is also discussed in variousmodified theories [53, 54, 57, 59, 62, 63].

We are interested to study the thermodynamic behav-ior in 𝑓(𝑅, 𝐿

𝑚) gravity which is a more general modified

theory with curvature matter coupling [55]. This theory canreproduce every action of non-minimal and arbitrary mattercurvature coupling in 𝑓(𝑅) gravity. In previous studies onthermodynamic properties in nonlinear theories, it has beenshown that non-equilibrium treatment is necessary in suchtype of theories [35, 36, 53, 54, 57, 59, 62, 63]. Here, weregard the non-equilibrium approach and show that the field

equations in 𝑓(𝑅, 𝐿𝑚) gravity can be cast to the form 𝑇𝑑(𝑆 +

𝑆) = −𝑑𝐸 + 𝑊𝑑𝑉, where 𝑑𝑆 is the entropy production term.Furthermore, we formulate the generalized second law ofthermodynamics (GSLT) and develop constraints by utilizingsome concrete examples in this theory.

The paper is organized as follows. In the next section,we formulate the field equations in 𝑓(𝑅, 𝐿

𝑚) gravity and

develop the first lawof thermodynamics (FLT) at the apparenthorizon of FRW universe. Section 3 investigates the validityof GSLT for some models in 𝑓(𝑅, 𝐿

𝑚) gravity. In Section 4,

we conclude our findings.

2. 𝑓(𝑅,L𝑚) Gravity and First

Law of Thermodynamics

The action of more general modified theory involving maxi-mal arbitrary curvature matter coupling is of the form [55]

A =1

𝜅2∫𝑓 (𝑅,L

𝑚)√−𝑔𝑑𝑥

4, (1)

where the function𝑓(𝑅,L𝑚)necessitates an arbitrary depen-

dence on scalar curvature 𝑅 and Lagrangian density L𝑚

which represents the matter contents. One can recover themodified 𝑓(𝑅) theories with matter curvature coupling fromaction (1). Consider 𝑓(𝑅, 𝐿

𝑚) = (1/2)𝑓

1(𝑅) + 𝐺(L

𝑚)𝑓2(𝑅),

where 𝑓𝑖and 𝐺(L

𝑚) are arbitrary functions of 𝑅 and 𝐿

𝑚,

respectively, it corresponds to 𝑓(𝑅) theory with arbitrarycurvature matter coupling. If we set 𝑓

1(𝑅) = 𝑓(𝑅), 𝐺(𝐿

𝑚) =

𝐿𝑚and 𝑓2(𝑅) = 1+𝜆𝑓

2(𝑅); then it implies the nonminimally

coupled𝑓(𝑅) gravity.Moreover, the corresponding actions inpure 𝑓(𝑅) and Einstein gravities can be reproduced by fixing𝑓1(𝑅) = 𝑓(𝑅), 𝐺(𝐿

𝑚) = 𝐿

𝑚, and 𝑓

2(𝑅) = 1 and 𝑓

1(𝑅) = 𝑅,

𝐺(𝐿𝑚) = 𝐿

𝑚, 𝑓2(𝑅) = 1, respectively. The matter energy-

momentum tensor is considered as follows:

𝑇(𝑚)𝜇] = −

2

√−𝑔

𝛿 (√−𝑔L𝑚)

𝛿𝑔𝜇], (2)

which implies that

𝑇(𝑚)𝜇] = 𝑔𝜇]L𝑚 −

2𝜕L𝑚

𝜕𝑔𝜇](3)

by assuming that the matter Lagrangian depends only uponthe metric tensor rather than on its derivatives.

The field equations in 𝑓(𝑅,L𝑚) gravity are

𝜅2

2𝑃 (𝑅, 𝐿

𝑚) 𝑇𝜇] = 𝐹 (𝑅, 𝐿𝑚) 𝑅𝜇]

−1

2[𝑓 (𝑅, 𝐿

𝑚) − 𝑃 (𝑅, 𝐿

𝑚) 𝐿𝑚] 𝑔𝜇]

− (∇𝜇∇] − 𝑔𝜇]◻)𝐹 (𝑅, 𝐿𝑚) ,

(4)

where we put 𝑃(𝑅, 𝐿𝑚) = 𝑓

𝐿𝑚

(𝑅, 𝐿𝑚) and 𝐹(𝑅, 𝐿

𝑚) =

𝑓𝑅(𝑅, 𝐿𝑚) so that representation of equations is more

convenient and subscripts 𝑅, 𝐿𝑚

point to the derivativeswith respect to scalar curvature and matter Lagrangian,

Advances in High Energy Physics 3

◻ = 𝑔𝛼𝛽∇𝛼∇𝛽, ∇𝜇represents covariant derivative related to

the Levi-Civita connection of the metric. One can retrievethe field equations in Einstein gravity just by replacing𝑓(𝑅, 𝐿

𝑚) = (𝑅/2) + 𝐿

𝑚. The contraction of (4) implies the

following relation:

𝜅2

2𝑃 (𝑅, 𝐿

𝑚) 𝑇 =𝐹 (𝑅, 𝐿

𝑚) 𝑅 − 2 [𝑓 (𝑅, 𝐿

𝑚) − 𝑃 (𝑅, 𝐿

𝑚) 𝐿𝑚]

+ 3◻𝐹 (𝑅, 𝐿𝑚) .

(5)

We can eliminate the term ◻𝑓𝑅(𝑅, 𝐿𝑚) from (4) and (5)

so that the field equations become

𝜅2

2𝑃 (𝑅, 𝐿

𝑚) (𝑇𝜇] −

1

3𝑇𝑔𝜇])

= 𝐹 (𝑅, 𝐿𝑚) (𝑅𝜇] −

1

3𝑅𝑔𝜇])

+1

6(𝑓 (𝑅, 𝐿

𝑚) − 𝑃 (𝑅, 𝐿

𝑚) 𝐿𝑚)

× 𝑔𝜇] − ∇𝜇∇]𝐹 (𝑅, 𝐿𝑚) .

(6)

Equation (4) can be sorted out to develop the form of theeffective Einstein field equation as

𝐺𝜇] = 𝑅𝜇] −

1

2𝑅𝑔𝜇] = 𝜅

2

eff𝑇(𝑚)

𝜇] + 𝑇(CM)𝜇] , (7)

where 𝜅eff = 8𝜋𝐺𝑃(𝑅, 𝐿𝑚)/2𝐹(𝑅, 𝐿𝑚) involves the effec-tive gravitational coupling and 𝑇(CM)

𝜇] denotes the energy-momentum tensor associated with curvaturematter couplingcomponents defined as

𝑇(CM)𝜇] =

1

𝐹 (𝑅, 𝐿𝑚)[1

2(𝑓 (𝑅, 𝐿

𝑚) − 𝑅𝐹 (𝑅, 𝐿

𝑚)) 𝑔𝜇]

+ (∇𝜇∇] − 𝑔𝜇]◻)

×𝐹 (𝑅, 𝐿𝑚) −

1

2𝑃 (𝑅, 𝐿

𝑚) 𝐿𝑚𝑔𝜇]] .

(8)

In this work, we assume the FRWmetric

𝑑𝑠2 = 𝑑𝑡2 − 𝑎2 (𝑡) (𝑑𝑟2

1 − 𝑘2𝑟2+ 𝑟2𝑑Ω2) ,

𝑑Ω2 = 𝑑𝜃2 + sin2𝜃 𝑑𝜙2(9)

which is necessarily homogenous as well as isotropic andperfect fluid is taken as matter energy-momentum tensor. Inthis framework, the 𝑓(𝑅, 𝐿

𝑚) field equations take the form

3 (𝐻2 +𝑘

𝑎2) = 𝜅2eff𝜌𝑀 + 𝜌CM, (10)

−2(�̇� −𝑘

𝑎2) = 𝜅2eff (𝜌𝑀 + 𝑝𝑀) + (𝜌CM + 𝑝CM) , (11)

where 𝜌𝑚and 𝑝

𝑚indicate the energy density and pressure

of matter fluid while 𝜌CM and 𝑝CM mark energy density andpressure of components produced due to matter geometrycoupling translated by the following expressions:

𝜌CM =1

𝑓𝑅

[1

2(𝑓 − 𝑅𝐹) − 3𝐻(𝐹

𝑅�̇� + 𝐹𝐿𝑚

�̇�𝑚) −

1

2𝑃𝐿𝑚] ,

𝑝CM =1

𝑓𝑅

[−1

2(𝑓 − 𝑅𝐹) + 2𝐻(𝐹


�̇�𝑚)

+ 𝐹𝑅�̈� + 𝐹𝑅𝑅

�̇�2 + 2�̇��̇�𝑚

×𝐹𝑅𝐿𝑚

+ 𝐹𝐿𝑚

�̈�𝑚+ 𝐹𝐿𝑚𝐿𝑚

(�̇�𝑚)2

+1

2𝑃𝐿𝑚] ,

(12)

where 𝐻 = ̇𝑎(𝑡)/𝑎(𝑡) is the Hubble parameter and dot insuperscript indicates time derivative. Consequently, the fieldequations in 𝑓(𝑅, 𝐿

𝑚) gravity can be organized as

−2(�̇� −𝑘

𝑎2) = 𝜅2eff (𝜌𝑚 + 𝑝𝑚)

+1

𝑓𝑅

[ − 𝐻(𝐹𝑅�̇� + 𝐹𝐿𝑚

�̇�𝑚) + 𝐹𝑅�̈� + �̇�2

× 𝐹𝑅𝑅

+ 2𝐹𝑅𝐿𝑚

�̇��̇�𝑚+ 𝐹𝐿𝑚

�̈�𝑚

+𝐹𝐿𝑚𝐿𝑚

(�̇�𝑚)2

] .

(13)

Now, we propose to formulate the FLT in this modifiedtheory at the apparent horizon of FRW universe. The condi-tion ℎ𝛼𝛽𝜕

𝛼𝑟𝜕𝛽𝑟 = 0 implies the radius of dynamical apparent

horizon for FRW geometry as

𝑟𝐴= (𝐻2 +

𝑘

𝑎2)−1/2

, (14)

which matches the Hubble horizon 𝑟𝐴

= 1/𝐻 for flat FRWuniverse. We differentiate (14) with respect to cosmic timewhich yields

1

𝐻𝑟3𝐴

𝑑𝑟𝐴

𝑑𝑡= (�̇� −

𝑘

𝑎2) . (15)

If we substitute (13) in the above relation andmultiply the restwith 4𝜋𝑟

𝐴, it follows that

1

2𝜋𝑟𝐴

(4𝜋𝑟𝐴𝐹

𝑃𝐺)𝑑𝑟𝐴

= 4𝜋𝑟3𝐴[𝜌𝑚+ 𝑝𝑚+

1

4𝜋𝐺𝑃

× ( − 𝐻(𝐹𝑅�̇� + 𝐹𝐿𝑚

�̇�𝑚)

+ 𝐹𝑅�̈� + 𝐹𝑅𝑅

�̇�2

+ 2𝐹𝑅𝐿𝑚

�̇��̇�𝑚+ 𝐹𝐿𝑚

�̈�𝑚


(�̇�𝑚)2

) ]𝐻𝑑𝑡,

(16)


which can be expressed as

1

2𝜋𝑟𝐴

𝑑 (𝐴𝐹

2𝑃𝐺)

= 4𝜋𝑟3𝐴[𝜌𝑚+ 𝑝𝑚+

1

4𝜋𝑃𝐺

× ( − 𝐻(𝐹𝑅�̇� + 𝐹𝐿𝑚

�̇�𝑚) + 𝐹𝑅�̈�

+ 𝐹𝑅𝑅

�̇�2 + 2𝐹𝑅𝐿𝑚

�̇��̇�𝑚+ 𝐹𝐿𝑚

�̈�𝑚


(�̇�𝑚)2

) ]𝐻𝑑𝑡

+𝑟𝐴

𝑃𝐺[𝐹𝑅�̇� + 𝐹𝐿𝑚

�̇�𝑚− 𝐹𝜕𝑡(ln𝑃)] 𝑑𝑡,

(17)

where 𝐴 = 4𝜋𝑟2𝐴is the area of apparent horizon. In (17), we

have employed the following differential:

𝑑(𝐴𝐹

2𝑃𝐺) =

4𝜋𝑟𝐴𝐹

𝑃𝐺𝑑𝑟𝐴− 2𝜋𝑟2

𝐴𝑑(

𝐹

𝑃𝐺) . (18)

Furthermore,multiplying the above equation by the term (1−( ̇̃𝑟𝐴/2𝐻𝑟𝐴)), we find

𝜅sg

2𝜋𝑑 (

𝐴𝐹

2𝑃𝐺) = (1 −

̇̃𝑟𝐴

2𝐻𝑟𝐴

)4𝜋𝑟3𝐴

× [𝜌𝑚+ 𝑝𝑚+

1

4𝜋𝑃𝐺

× ( − 𝐻(𝐹𝑅�̇� + 𝐹𝐿𝑚

�̇�𝑚)

+ 𝐹𝑅�̈� + 𝐹𝑅𝑅

�̇�2 + 2𝐹𝑅𝐿𝑚

�̇��̇�𝑚

+𝐹𝐿𝑚


(�̇�𝑚)2

) ]𝐻

× 𝑑𝑡 + (1 −̇̃𝑟𝐴

2𝐻𝑟𝐴

)𝑟𝐴

𝑃𝐺

× [𝐹𝑅�̇� + 𝐹𝐿𝑚

�̇�𝑚− 𝐹𝜕𝑡(ln𝑃)] 𝑑𝑡,

(19)

where 𝑇 = |𝜅sg|/2𝜋 is the temperature of apparent hori-zon which includes the surface gravity 𝜅sg = (1/𝑟𝐴)(1 −( ̇̃𝑟𝐴/2𝐻𝑟𝐴)).

The Bekenstein-Hawking relation [40–43] 𝑆 = 𝐴/4𝐺defines the horizon entropy in Einstein-gravity. In alternativetheories of gravity, Wald [56] suggested that horizon entropyis associated with a Noether charge and in 𝑓(𝑅) gravity,entropy is defined as 𝑆 = 𝐴𝐹/4𝐺 [53, 54, 57]. Bamba andGeng [53, 54, 57] pointed out that Wald entropy relationis similar for both metric and Palatini formalisms in 𝑓(𝑅)theory. Brustein et al. [58] demonstrated that Wald entropyis equivalent to 𝑆 = 𝐴/4𝐺eff, where 𝐺eff being the effectivegravitational coupling. Therefore, the Wald entropy at theapparent horizon in 𝑓(𝑅, 𝐿

𝑚) gravity can be defined as

[35, 36, 53, 54, 57, 59] 𝑆 = 𝐴𝐹/2𝑃𝐺 which reproduces

the corresponding result in 𝑓(𝑅) theory involving arbitrarycurvature matter coupling [59] for 𝑓(𝑅, 𝐿

𝑚) = (1/2)𝑓

1(𝑅) +

𝑔(L𝑚)𝑓2(𝑅). Consequently, (19) can be rewritten as

𝑇𝑑𝑆 = 4𝜋𝑟3𝐴(𝜌𝑚+ 𝑝𝑚)𝐻𝑑𝑡 − 2𝜋𝑟2

𝐴(𝜌𝑚+ 𝑝𝑚) 𝑑𝑟𝐴

+𝑇𝐴

2𝑃𝐺[𝑟2𝐴( − 𝐻(𝐹


�̇�𝑚)

+ 𝐹𝑅�̈� + 𝐹𝑅𝑅

�̇�2 + 2𝐹𝑅𝐿𝑚

�̇��̇�𝑚

+𝐹𝐿𝑚


× (�̇�𝑚)2

)𝐻

+𝐹𝑅�̇� + 𝐹𝐿𝑚

�̇�𝑚− 𝐹𝜕𝑡(ln𝑃) ] 𝑑𝑡.

(20)

Now, we pay attention to matter energy inside a sphereof radius 𝑟

𝐴at apparent horizon, that is, 𝐸 = 𝑉𝜌, where

𝑉 = 4/3𝜋𝑟3𝐴is the volume of 3-dimensional sphere. The time

derivative of energy relation gives

𝑑𝐸 = 4𝜋𝑟2𝐴𝜌𝑑𝑟𝐴+ 𝑉 ̇𝜌𝑑𝑡. (21)

If one considers the covariant divergence of (4), then aftersome manipulations it leads to

∇𝛼𝑇𝛼𝛽

= 2∇𝛼 ln [𝑃 (𝑅, 𝐿𝑚)]

𝜕𝐿𝑚

𝜕𝑔𝛼𝛽, (22)

which implies that the usual continuity equation does nothold in this modified theory which is similar to other mod-ified gravities involving matter geometry coupling. The cor-responding results in 𝑓(𝑅) theory with arbitrary and non-minimal matter geometry coupling can be reproduced forparticular choices of Lagrangian.

In FRW framework, the energy density meets the follow-ing equation

̇𝜌 + 3𝐻 (𝜌 + 𝑝) = 2𝜕𝑡[𝑃 (𝑅, 𝐿

𝑚)]

𝜕𝐿𝑚

𝜕𝑔𝜇]

𝑔00. (23)

Using this equation in (21), we obtain

𝑑𝐸 = 4𝜋𝑟2𝐴𝜌𝑑𝑟𝐴− 3𝑉 (𝜌 + 𝑝)𝐻𝑑𝑡 + 2𝑉𝜕

𝑡[ln𝑃]

𝜕𝐿𝑚

𝜕𝑔𝜇]

𝑔00𝑑𝑡.

(24)

Hence, (20) can be represented as

𝑇ℎ𝑑𝑆ℎ= − 𝑑𝐸 + 𝑊𝑑𝑉 +

𝑇𝐴

2𝑃𝐺

× [𝑟2𝐴𝐻( − 𝐻(𝐹


�̇�𝑚) + 𝐹𝑅�̈�

+ 𝐹𝑅𝑅

�̇�2 + 2𝐹𝑅𝐿𝑚

�̇��̇�𝑚+ 𝐹𝐿𝑚

�̈�𝑚


(�̇�𝑚)2

) + 𝐹𝑅�̇� + 𝐹𝐿𝑚

�̇�𝑚

−𝐹𝜕𝑡(ln𝑃) ] 𝑑𝑡

+ 𝑇4𝜋𝑉

|𝜅|𝜕𝑡(ln𝑃)

𝜕𝐿𝑚

𝜕𝑔𝜇]

𝑔00𝑑𝑡,

(25)


where 𝑊 = (1/2)(𝜌 − 𝑝) is the work density. Equation (25)shows that the field equations in this theory does not obey thestandard form of FLT, 𝑇𝑑𝑆 = −𝑑𝐸 + 𝑊𝑑𝑉 which is satisfiedin Einstein, Lovelock, and Gauss-Bonnet gravities.The aboverelation involves additional terms which are produced dueto the non-equilibrium representation of thermodynamics.Thus the FLT in this theory can be represented as

𝑇ℎ𝑑𝑆ℎ+ 𝑇ℎ𝑑𝑆ℎ= −𝑑𝐸 + 𝑊𝑑𝑉, (26)

where

𝑑𝑆ℎ=

−𝐴

2𝑃𝐺[𝑟2𝐴𝐻( − 𝐻(𝐹


�̇�𝑚)

+ 𝐹𝑅�̈� + 𝐹𝑅𝑅

�̇�2 + 2𝐹𝑅𝐿𝑚

�̇��̇�𝑚

+𝐹𝐿𝑚


(�̇�𝑚)2

)

+𝐹𝑅�̇� + 𝐹𝐿𝑚

�̇�𝑚− 𝐹𝜕𝑡(ln𝑃) ] 𝑑𝑡

−4𝜋𝑉

|𝜅|𝜕𝑡(ln𝑃)

𝜕𝐿𝑚

𝜕𝑔𝜇]

𝑔00𝑑𝑡

(27)

denotes the entropy production term which is a result ofmatter geometry coupling inmore general𝑓(𝑅) gravity at theapparent horizon of FRW universe.

One can retrieve 𝑑𝑆 in pure 𝑓(𝑅) theory for the Lagran-gian 𝑓(𝑅, 𝐿

𝑚) = (1/2)𝑓(𝑅) + 𝐿

𝑚which is identical with

the results in [53, 54, 57]. It is remarked that we establishthe FLT in more general modified gravity which recoversthe corresponding laws in 𝑓(𝑅) theory involving arbitrary aswell as non-minimal gravitational coupling. For 𝑓(𝑅, 𝐿

𝑚) =

(1/2)𝑓1(𝑅) + 𝑔(L

𝑚)𝑓2(𝑅), we can get the respective FLT

in 𝑓(𝑅) theory having arbitrary coupling which is alsoformulated in [59]. If one defines effective entropy as a sumof horizon entropy and entropy production term 𝑆eff = 𝑆ℎ+𝑆,then FLT can also be represented in the form𝑇

ℎ𝑑𝑆eff = −𝑑𝐸+

𝑊𝑑𝑉.

3. GSLT in 𝑓(𝑅, 𝐿𝑚) Gravity

Here, we investigate the validness of GSLT in the context of𝑓(𝑅, 𝐿

𝑚) gravity at the apparent horizon. It states that the

sum of the horizon entropy and entropy of ordinary matterfluid components always increases in time. The validity ofGSLT has been tested in modified gravities including 𝑓(𝑅),𝑓(T), 𝑓(𝑅, 𝑇), and 𝑓(𝑅) theories involving matter geometrycoupling. It would be interesting to examine the GSLT in thismodified theory which can reproduce other curvature mattercoupled modified gravities. The entropy associated with theapparent horizon is given by 𝑆

ℎ= 𝐴𝐹/2𝑃𝐺 and its time

derivative is

̇𝑆ℎ=

2𝜋𝑟𝐴

𝑃𝐺[𝑟𝐴(�̇� − 𝐹

𝑑

𝑑𝑡ln [𝑃]) + 2 ̇̃𝑟

𝐴𝐹] . (28)

The evolution of 𝑆ℎmultiplied with 𝑇

ℎcan be found as

𝑇ℎ

̇𝑆ℎ=

1

𝑃𝐺(1 −

̇̃𝑟𝐴

2𝐻𝑟𝐴

)[𝑟𝐴(�̇� − 𝐹

𝑑

𝑑𝑡ln [𝑃]) + 2 ̇̃𝑟

𝐴𝐹] .

(29)

The dynamical equation relating the entropy of mattersources inside the horizon 𝑆in and temperature 𝑇in to thedensity and pressure in the horizon is given by

𝑇in𝑑𝑆in = 𝑑 (𝜌𝑚𝑉) + 𝑝𝑚𝑑𝑉. (30)

The evolution of entropy inside the horizon can be foundusing (23) as

𝑇in ̇𝑆in = 4𝜋𝑟2

𝐴(𝜌𝑚+ 𝑝𝑚) ( ̇̃𝑟𝐴− 𝐻𝑟𝐴)

+ 2𝑉𝜕𝑡(ln [𝑃])

𝜕𝐿𝑚

𝜕𝑔𝜇]

𝑔00,

(31)

where matter energy density and pressure can be evaluatedfrom (4) which yields for FRW spacetime as

𝜌𝑚

=2

𝜅2𝑃[1

2(𝑃𝐿𝑚− 𝑓) − 3 (�̇� + 𝐻2) 𝐹 + 3𝐻�̇�] ,

𝑝𝑚

=2

𝜅2𝑃[1

2(𝑓 − 𝑃𝐿

𝑚) + (�̇� + 3𝐻2 +

2𝜅2

𝑎2)𝐹

− 2𝐻�̇� − �̈�] .

(32)

Substituting 𝜌𝑚and 𝑝

𝑚in (31), we obtain

𝑇in ̇𝑆in = 8𝜋𝑟2

𝐴( ̇̃𝑟𝐴− 𝐻𝑟𝐴)

2

𝜅2𝑃[2 (

𝜅

𝑎2− �̇�)𝐹 + 𝐻�̇� − �̈�]

+ 2𝑉𝜕𝑡(ln𝑃)

𝜕𝐿𝑚

𝜕𝑔𝜇]

𝑔00.

(33)

Now, we proceed to establish the GSLT in this modifiedtheory which requires (�̇�

ℎ

̇𝑆ℎ+�̇�in ̇𝑆in) ⩾ 0.The temperature of

matter and energy components inside the horizon is assumedin proportion to temperature of apparent horizon, that is,𝑇in = 𝑏𝑇ℎ, where 0 < 𝑏 < 1. In fact, it is natural toconsider such proportionality relation which results in localequilibrium by setting the proportionality constant 𝑏 as unity.In general, the horizon temperature does not match to thatof fluid components inside the horizon and this differencemakes the spontaneous flow of energy between the horizonand fluid contents so that local thermal equilibrium is nolonger preserved [60].Moreover, themutualmatter geometrycoupling in this theory may also play its role in energy flowand systems must go through some interaction for someperiod of time before achieving the thermal equilibrium.Thus, the GSLT requires 𝑇

ℎ

̇𝑆tot = �̇�ℎ( ̇𝑆ℎ + ̇𝑆in) ⩾ 0, using (29)and (33), we have

𝑇ℎ

̇𝑆tot =1

2𝑃𝐺(𝐻2 +

𝜅

𝑎2)−5/2

× [2𝐻(𝜅

𝑎2− �̇�) {

𝜅

𝑎2+ (1 − 2𝑏) �̇�

+ 2 (1 − 𝑏) × 𝐻2}𝐹

− (𝜅

𝑎2+ 𝐻2)(

𝜅

𝑎2+ �̇� + 2𝐻2) 𝜕

𝑡(ln𝑃) 𝐹


+ {𝜅

𝑎2(𝜅

𝑎2+ �̇� + 3𝐻2) + (1 − 2𝑏) �̇�𝐻

2

+ 2 (1 − 𝑏)𝐻4} (𝐹𝑅�̇� + 𝐹𝐿𝑚

�̇�𝑚)

+ 2𝑏𝐻 (�̇� + 𝐻2)

× {𝐹𝑅�̈� + 𝐹𝑅𝑅

�̇�2 + 2𝐹𝑅𝐿𝑚

�̇��̇�𝑚

+𝐹𝐿𝑚


× (�̇�𝑚)2

}

+16𝜋𝑏𝑃𝐺

3(𝜅

𝑎2+ 𝐻2) 𝜕

𝑡(ln𝑃)

×𝜕𝐿𝑚

𝜕𝑔𝜇]

𝑔00] ⩾ 0,

(34)

which is a constraint to meet GSLT in this modified theoryand it counts on the basis of different choices in Lagrangian.This condition is more comprehensive and one can deducethe corresponding results in Einstein, 𝑓(𝑅), and 𝑓(𝑅) the-ories having non-minimal and arbitrary curvature mattercoupling. For 𝑓(𝑅, 𝐿

𝑚) = (1/2)𝑓

1(𝑅) + 𝑔(L

𝑚)𝑓2(𝑅), we get

the GSLT in 𝑓(𝑅) gravity with arbitrary matter geometrycoupling and if 𝑔(𝐿

𝑚) = 𝐿

𝑚, 𝑓2(𝑅) = 1 + 𝜆𝑔(𝑅), then

the corresponding result in 𝑓(𝑅) gravity with non-minimalcoupling can be found [59].

In this setting, we limit our discussion to hypothesisof thermal equilibrium so that energy would not flow inthe system and horizon temperature is more or less equalto temperature inside the horizon. This situation wouldcorrespond to the case of late times where universe compo-nents and horizon would have interacted for long time [61]while its existence for early or intermediate times would beambiguous. Though the assumption of thermal equilibriumis limiting in some sense to avoid the non-equilibriumcomplexities but it has widely been accepted to study theGSLT [53, 54, 57, 62, 63]. In [35, 36], we have discussedthe GSLT under non-equilibrium picture of thermodynamicsin 𝑓(𝑅, 𝑇) gravity. Here, we consider the case of thermalequilibrium with 𝑏 = 1 so that horizon temperature is equalto that of fluid components inside the horizon.

To illustrate the validity of GSLT in 𝑓(𝑅, 𝐿𝑚) gravity, we

take concrete model in this modified theory [59]

𝑓 (𝑅, 𝐿𝑚) = 𝜆 exp( 1

2𝜆𝑅 +

1

𝜆𝐿𝑚) , (35)

where 𝜆 > 0 is an arbitrary constant, and if (𝑅/2𝜆) +(𝐿𝑚/𝜆) ≪ 1, then 𝑓(𝑅, 𝑇) ≈ 𝜆 + 𝑅/2 + 𝐿

𝑚+ ⋅ ⋅ ⋅ represents

the ΛCDMmodel. Substituting model (35) in (34), it impliesthat

𝑇ℎ

̇𝑆tot =1

4𝐺(𝐻2 +

𝜅

𝑎2)−5/2

× [2𝐻(𝜅

𝑎2− �̇�)2

− (𝜅

𝑎2+ 𝐻2)

× (𝜅

𝑎2+ �̇� − 2𝐻2) 𝜕

𝑡(

1

2𝜆(𝑅 + 2𝐿

𝑚))

+ {𝜅

𝑎2(𝜅

𝑎2+ �̇� + 3𝐻2) − �̇�𝐻2}

× 𝜕𝑡(

1

2𝜆(𝑅 + 2𝐿

𝑚)) + 2𝐻(�̇� + 𝐻2)

× (exp ( 12𝜆

(𝑅 + 2𝐿𝑚)))−1

× 𝜕𝑡𝑡exp ( 1

2𝜆(𝑅 + 2𝐿

𝑚)) +

32𝜋𝐺

3(𝜅

𝑎2+ 𝐻2)

× 𝜕𝑡(

1

2𝜆(𝑅 + 2𝐿

𝑚)) ×

𝜕𝐿𝑚

𝜕𝑔𝜇]

𝑔00] ⩾ 0,

(36)

which is a constraint to validate GSLT for model (35). To bemore explicit about the above constraint, we choose 𝐿

𝑚=

𝜌 and set power law cosmology 𝑎(𝑡) = 𝑎0𝑡𝑚, (𝑚 > 1) with

𝜌 = 𝜌0𝑡−3𝑚. We plot GSLT for flat FRW geometry shown in

Figure 1. To getmore insights of these conditions, we considerthe model of the form 𝑓(𝑅, 𝐿

𝑚) = 𝛼𝑅 + 𝛽𝑅2 + 𝛾𝐿

𝑚, where 𝛼,

𝛽 and 𝛾 are arbitrary constants, and hence the GSLT becomes

𝑇ℎ

̇𝑆tot =1

2𝛾𝐺(𝐻2 +

𝜅

𝑎2)−5/2

× [2𝐻(𝜅

𝑎2− �̇�)2

(𝛼 + 2𝛽𝑅)

+ 2𝛽 {𝜅

𝑎2(𝜅

𝑎2+ �̇� − 3𝐻2) − �̇�𝐻2} �̇�

+ 4𝛽𝐻(�̇� + 𝐻2) �̈�] ⩾ 0.

(37)

For the flat FRWuniverse, the above condition depends uponthe parameters 𝛼, 𝛽, and 𝛾 which can be satisfied if 𝛽 < 0 and(𝛼, 𝛾) > 0.The validity ofGSLT for the secondmodel is shownin Figure 2.

4. Conclusion

In this paper, we have discussed thermodynamic propertiesin more general modified gravity characterized by curvaturematter coupling. We have found that Friedmann equationscan be cast to the fundamental form of FLT 𝑇

ℎ𝑑𝑆eff = 𝛿𝑄,

where 𝑇ℎis the horizon temperature and 𝑆eff is the effective

entropy which consists of two factors 𝑆eff = 𝑆ℎ + 𝑆, the firstdenotes the horizon entropy satisfying the usual FLT andother factor is the entropy production term. It is remarkedthat entropy production term appears due to the non-equilibrium description in 𝑓(𝑅, 𝐿

𝑚) gravity. This indicates

that one may need non-equilibrium treatment of thermody-namics in this theory. The entropy production term appearsto be more general which can reproduce the correspondingresults in Einstein, pure𝑓(𝑅) and𝑓(𝑅) gravity involving non-minimal and arbitrary curvature matter coupling.


0

5

10t

2

4

6

8

10

0204060

𝜆

Tḣ Stot

(a)

0

5

102

4

6

8

10

0

50

Tḣ Stot

t

m

(b)

Figure 1: Evolution of GSLT for the Lagrangian 𝑓(𝑅, 𝐿𝑚) = 𝜆 exp((1/2𝜆)𝑅 + (1/𝜆)𝐿

𝑚), plot (a) shows the bound on parameter 𝜆 for𝑚 = 10,

whereas in plot (b) we set 𝜆 = 2 to find bounds on𝑚. It is evident that GSLT is valid if 𝜆 > 0,𝑚 > 1.

0

5

102

4

6

8

10

050

100

150

Tḣ Stot

t

m

(a)

0

5

10

0

5

100

100

Tḣ Stot

t

𝛽

−10

−5

−100

(b)

Figure 2: Evolution of GSLT for 𝑓(𝑅, 𝐿𝑚) = 𝛼𝑅 + 𝛽𝑅2 + 𝛾𝐿

𝑚, plot (a) shows the bound on parameter 𝑚 for 𝛽 = −5, whereas in plot (b) we

set𝑚 = 10 to find bounds on 𝛽. We choose 𝛼 = 𝛾 = 2.

Moreover, in modified theories with explicit matter grav-ity coupling, we have nondiminishing covariant divergenceof the energy-momentum tensor which demonstrates theenergy flow due to mutual interaction of matter and gravity.The non-equilibrium treatment can be translated due toenergy flow on the apparent horizon of FRW universe. Wehave also examined the time evolution of total entropyincluding the horizon entropy and entropy inside the hori-zon. The proportionality relation between temperature ofapparent horizon and temperature associated with mattercontents inside the horizon is assumed to develop the GSLT.We have also found constraints on two specific gravita-tional models 𝑓(𝑅, 𝐿

𝑚) = 𝜆 exp((1/2𝜆)𝑅 + (1/𝜆)𝐿

𝑚) and

𝑓(𝑅, 𝐿𝑚) = 𝛼𝑅+𝛽𝑅2+𝛾𝐿

𝑚to secure the GSLT in this theory.

We have checked the validity of GSLT for flat FRW geometrywith local thermal equilibrium and constrained the coupling

parameters. For the first model, GSLT is satisfied if 𝜆 > 0 and𝑚 > 1, whereas for the second case it requires 𝛽 < 0 and(𝛼, 𝛾) > 0.

Acknowledgment

Theauthors thank theHigher EducationCommission, Islam-abad, Pakistanm, for its financial support through the Indige-nous Ph.D. 5000 Fellowship Program Batch-VII.

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