Thermodynamics in <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-4.5269pt" id="M1" height="16.7875pt" version="1.1" viewBox="-0.0657574 -12.2606 34.0807 16.7875" width="34.0807pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-103" d="M619 670C619 686 593 712 555 712S459 686 410 634S335 504 320 430H250L219 400L222 388H312L258 73C223 -133 201 -166 187 -180C175 -191 158 -199 140 -199C123 -199 88 -188 74 -172C68 -166 63 -164 54 -171C38 -185 23 -201 23 -215C23 -236 60 -261 93 -261C122 -261 161 -247 207 -200C268 -138 300 -71 337 94C365 220 376 277 394 387L501 399L521 430H401C432 623 464 665 501 665C524 665 544 651 567 627C577 617 583 618 592 625C601 631 619 651 619 670Z"/></g><g transform="matrix(.017,0,0,-0.017,11.025,0)"><path id="g113-41" d="M300 -147C201 -63 143 98 143 270S200 602 300 686L282 710C136 610 70 450 70 271V270C70 89 136 -72 282 -170L300 -147Z"/></g><g transform="matrix(.017,0,0,-0.017,16.962,0)"><path id="g113-85" d="M620 675H597C578 656 570 650 541 650H144C112 650 104 653 94 675H72C59 618 42 552 23 493L53 491C71 534 88 564 105 585C124 608 144 615 238 615H290L197 121C182 40 174 34 88 28L82 0H361L367 28C275 34 266 38 281 121L374 615H441C522 615 543 608 553 583C562 560 566 531 565 493L597 494C603 551 612 629 620 675Z"/></g><g transform="matrix(.017,0,0,-0.017,27.898,0)"><path id="g113-42" d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z"/></g></svg> Gravity with Nonminimal Coupling to Matter

Azizi, Tahereh; Borhani, Najibeh

doi:https://doi.org/10.1155/2017/6839050

Advances in High Energy Physics

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Research Article | Open Access

Volume 2017 | Article ID 6839050 | https://doi.org/10.1155/2017/6839050

Thermodynamics in Gravity with Nonminimal Coupling to Matter

Tahereh Azizi¹and Najibeh Borhani¹

Academic Editor: Chao-Qiang Geng

Received17 Sept 2016

Revised28 Nov 2016

Accepted28 Dec 2016

Published17 Jan 2017

Abstract

In the present paper, we study the thermodynamics behavior of the field equations for the generalized gravity with arbitrary coupling between matter and the torsion scalar. In this regard, we explore the verification of the first law of thermodynamics at the apparent horizon of the Friedmann-Robertson-Walker universe in two different perspectives, namely, the nonequilibrium and equilibrium descriptions of thermodynamics. Furthermore, we investigate the validity of the second law of thermodynamics for both descriptions of this scenario with the assumption that the temperature of matter inside the horizon is similar to that of horizon.

1. Introduction

The teleparallel equivalent of general relativity (TEGR) [1, 2] is an equivalent formulation of classical gravity, which, instead of using the curvature defined via the Levi-Civita connection, uses the Weitzenböck connection that has no curvature but only torsion. This approach is closely related to the standard general relativity, differing only in “boundary terms” involving total derivatives in the action. In this setup, the dynamical objects are the four linearly independent vierbeins and the Lagrangian density, , is constructed from the torsion tensor which is formed solely from products of the first derivatives of the vierbein [2]. However, in a similar manner to the modified gravity [3–7], the teleparallel gravity is generalized to a modified version [8–11]. The Lagrangian density is an arbitrary function of the torsion scalar . This modification enables the theory to explain the late-time acceleration of the universe [9, 12, 13] which is favored by the observational data. So there is no need to introduce a mysterious dark energy component for the matter content of the universe. The significant advantage of gravity is that the field equations are second-order differential equations and are more manageable compared to theories. For some gravitational and cosmological aspects of the modified teleparallel gravity, see [14]. Recently, further generalization of the teleparallel gravity has been introduced in [15] by considering nonminimal coupling between matter and the torsion scalar in the action. In this model, the gravitational field can be described in terms of two arbitrary functions of the torsion scalar , namely, and , with the function linearly coupled to the matter Lagrangian [15]. This nonminimal torsion-matter coupling scenario can offer a unified description of the universe evolution from its inflationary to the late-time accelerated phases [15]. In [16], the energy conditions of this model are studied and the validity of energy bounds is examined. The dynamical system analysis for the cosmological applications of this model is carried out in [17].

In the present work, we are going to study the thermodynamics aspects of this nonminimally coupled model at the apparent horizon of an expanding cosmological background. Indeed, the black hole thermodynamics sets up connections between general relativity and the laws of thermodynamics [18]. In this content, a temperature and entropy, which are proportional to the surface gravity and area of the horizon, respectively, are associated with the black hole. The first law of black hole thermodynamics is given by the identity [19], where is the mass of the black hole. Furthermore, Jacobson [20] showed that Einstein’s equations can be derived from the fundamental relation in all local Rindler horizons, where and are the energy flux across the horizon and Unruh temperature, respectively. This approach soon generalized to the cosmological situation, where it was shown that, by applying the Clausius relation to the apparent horizon of the Friedmann-Robertson-Walker (FRW) universe, the Friedmann equation can be rewritten in the form of the first law of thermodynamics [21]. Recently, the equivalence of the Clausius relation and the gravitational field equations has been investigated to the more general modified theories of gravity such as Gauss-Bonnet gravity [22], Lovelock gravity [23, 24], Braneworld gravity [25], scalar-tensor gravity [26], theories [27–31], and the extended models of -gravity [32–37]. In the context of gravity, the first law of black hole thermodynamics has been studied in [38] and the thermodynamics of the apparent horizon of the FRW universe is explored in [39–41]. This issue is also studied in some modified scenarios [42–45].

On the other hand, in [27], it was pointed out that, in order to derive the field equations of modified gravity, one should employ a nonequilibrium thermodynamics treatment. However, it has been demonstrated in [29] that it is possible to obtain an equilibrium description of thermodynamics on the apparent horizon of gravity. The same works also have been carried out in gravity [40] and some extended models of gravity [29, 37]. In addition to the first law of thermodynamics, there has been a lot of interest in exploring the second law of thermodynamics in gravitational theories [46–54]. According to the second law of thermodynamics, the sum of the horizon entropy and the entropy of the matter field, that is, the total entropy, is a nondecreasing function of time.

In this paper, we explore the laws of thermodynamics in both nonequilibrium and equilibrium descriptions in the nonminimal gravity model. The organization of the paper is as follows. In Section 2, we briefly review the nonminimally torsion-matter coupling model and its equations of motion. In Section 3, we treat nonequilibrium descriptions of thermodynamics and investigate the first and second laws of thermodynamics. We explore the equilibrium description of thermodynamics in Section 4. Finally, our conclusion will appear in Section 5.

2. The Equations of Motion

In the context of the teleparallel gravity, the dynamical object is a vierbein field , , which is an orthonormal basis for the tangent space at each point of the manifold. The metric tensor is obtained from the dual vierbein as , where is the Minkowski metric and is the component of the vector in a coordinate basis. Note that the Greek indices label coordinates on the manifold, while Latin indices refer to the tangent space. The torsion tensor is defined asDefining other two tensors,one can write down the torsion scalar . Using the torsion scalar as the teleparallel Lagrangian leads to the same gravitational equations of the general relativity. In this work, we focus on the modified teleparallel gravity with nonminimal coupling between the torsion scalar and the matter Lagrangian which is introduced via the following action [15]:where , and are arbitrary functions of the torsion scalar, and is a coupling constant with units of . Varying the action with respect to the vierbein leads to the field equations [15]:where and the prime denotes a derivative with respect to the torsion scalar and we have defined . We assume that the matter content of the universe is given by a perfect fluid and the matter Lagrangian density is described by which leads to . So the energy momentum tensor of the matter is given bywhere is the four velocities of the fluid in the comoving coordinates. For a flat homogeneous and isotropic Friedman-Robertson-Walker (FRW) universe, the vierbein is given bywhere is the cosmological scale factor. Using the above relation together with (1) and (2), one obtains , where is the Hubble parameter. The substitution of the FRW vierbein (6) in the field equation (4) yields the modified Friedmann equations as follows:where and . Note that the usual gravity can be recovered in the limit . It has been shown that (7) can describe the acceleration expansion of the universe without introduction of any dark energy component [15]. In the rest of this paper, we concentrate on the thermodynamic aspects of the nonminimal torsion-matter coupling extension of teleparallel gravity.

3. Nonequilibrium Picture

To study the thermodynamics of the nonminimal gravity, we rewrite (7) as follows:where the energy density and pressure of the dark components are defined asrespectively. Here, a hat denotes quantities in the nonequilibrium description of thermodynamics which do not satisfy the standard continuity equation, so thatThe perfect fluid satisfies the continuity equations by virtue of the Bianchi identity:

3.1. First Law of Thermodynamics

Now we investigate the thermodynamic behavior of the nonminimal gravity on the apparent horizon. In the flat FRW universe, the radius of the dynamical apparent horizon is given by [22]By taking the time derivative of this equation and substituting (10) into the result, we obtainIn the Einstein gravity, the Bekenstein-Hawking relation defines the horizon entropy, where is the area of the apparent horizon [55]. In the framework of the generalized theories of gravity such as modified gravity, a horizon entropy associated with a Noether charge, called the Wald entropy [56], is expressed as , where is the effective gravitational coupling [57].

In the context of gravity, it has been shown that the first law of black hole thermodynamics breaks down [38] due to the violation of local Lorentz invariance [58]. However, it is argued that when is small, the entropy of the black hole in gravity is approximately equal to . Furthermore, from the study of the matter density perturbations in gravity, one can take the effective gravitational coupling taken as . Hence, similar to case, with the Friedmann equation (10), we take the effective gravitational coupling as , so the Wald entropy in nonminimal gravity is given by

Using (16) and (17), we findThe temperature of the apparent horizon is given by the Hawking temperature , where is the surface gravity at the apparent horizon. Multiplying (18) with the term yieldsIn the Einstein gravity, the total energy inside a sphere of radius of the apparent horizon is . However, in the context of generalized gravity, one should use the effective gravitational constant in this relation. Hence, in the nonminimal modified gravity, the total energy is given by the following equation:where is the volume of 3-dimensional sphere. Taking the time derivative of (20), we find Using (19) and (21) leads toBy introducing the work density [59], one can rewrite (22) as follows:The above equation consists of additional term which is produced due to the nonequilibrium representation of thermodynamics. Consequently, the first law of thermodynamics can be expressed as follows:where we have defined an entropy production term asThe appearance of the additional term illustrates that the horizon thermodynamics is nonequilibrium one in the case of nonminimal gravity. Indeed, the violation of the standard first law of thermodynamics in this case is a result of the definition of the dark energy momentum components as and , which do not satisfy the continuity equation (13). In the next section, we show that, by definition of the energy density and pressure of this generalised scenario in a way that the new components satisfy the continuity equation, it is possible to have an equilibrium description of thermodynamics, so the first law of thermodynamics can be justified.

3.2. Second Law of Thermodynamics

In this subsection, we investigate the validity of the second law of thermodynamics at the apparent horizon in the framework of the nonminimal torsion-matter coupling model. The second law of thermodynamics states that the sum of the horizon entropy and the entropy of the matter field, that is, the total entropy, is a nondecreasing function of time. Assuming same temperature between the outside and inside of the apparent horizon, the condition to satisfy the second law of thermodynamics is given bywhere and are deduced from the first law of thermodynamics (see (24)) and can be extracted from Gibb’s equation which relates the entropy of all matter and energy sources to the pressure in the horizon, so thatwhere and are corresponding to the temperature and entropy of total energy inside the horizon, respectively, and we have defined and . Taking the time derivative of (27) and using (13) and (14), one can getNow, substituting the above equation and (24) in (26), we findThis result describes the validity of the second law of thermodynamics in the nonequilibrium treatment. So the condition needed to hold the second law of thermodynamics in nonminimal gravity is equivalent to . Note that should be positive in order to . This condition imposes a constraint to the coupling parameter , so that . As a result, the upper bound of lambda depends explicitly on the choices of two functions and . In the flat FRW universe, the effective equation of state parameter is defined as [5]Here () represents the quintessence phase of the universe, while () is corresponding to the phantom phase. From (26), we find that, in the nonequilibrium picture, the second law of thermodynamics is satisfied in both phantom and quintessence phases of the universe evolution.

4. Equilibrium Picture

In this section, we investigate the possibility to have an equilibrium picture of thermodynamics in the nonminimal modified gravity setup. To do this, we rewrite the Friedmann equations (7) in the following form:Equations (31) and (32) can be expressed aswhere we have defined the energy density and pressure of dark components asrespectively. Now, from the new definition of the dark energy density and pressure, the standard continuity equation can be retrieved as follows:So, the equilibrium description of thermodynamics can be treated in the same manner as general relativity.

4.1. First Law of Thermodynamics

In the new representation of the dark energy components, the time derivative of the dynamically apparent horizon is given byIntroducing the Bekenstein-Hawking entropy as , we findSubstituting the horizon temperature () in the above equation leads toDefining the Misner-Sharp energy as in the Einstein gravity, we getWith the definition of the work density as [59] and using (40) and (41), we obtain the following equation corresponding to the first law of equilibrium thermodynamics:As a result, an equilibrium treatment of the first law of thermodynamics is obtained with a suitable definition for the energy momentum tensor components of dark energy.

One can show that the horizon entropy in the equilibrium picture has a relation with the horizon entropy in the nonequilibrium picture asUsing (25) and (39), the above relation reduces to the following form:The difference between and appearing in the nonminimal gravity is due to . Note that, in the pure teleparallel gravity case, which corresponds to and , we have . From (44), we see that the change of the entropy in the equilibrium picture involves the information of both and in the nonequilibrium picture.

4.2. Second Law of Thermodynamics

In this subsection, we use the first law of thermodynamics in the equilibrium picture (see (42)) to find the general condition which is needed to hold the second law of thermodynamics in the nonminimal scenario. As before, we assume the same temperature for outside and inside the apparent horizon of the universe. To examine the validity of the second law of thermodynamics, we consider the Gibbs relation in terms of all matter and energy components:where and . Using (32) and (40), the evolution of the horizon entropy is given byTo obey the second law of thermodynamics in the equilibrium picture of nonminimal gravity, we require that . Plugging (45) and (46) into this condition, we findIt is clear that this condition naturally holds in the phantom phase of the universe, where . On the other hand, for the validity of the second law in the quintessence phase (), it is required that . Note that the result (47) is independent of the forms of two functions and imposes no constraint to the coupling parameter .

5. Conclusions

In this paper, we have studied the laws of thermodynamics in a generalized teleparallel gravity which contains nonminimal coupling between the matter field and an arbitrary function of the torsion scalar. From the Friedmann equations, we have constructed the first law of thermodynamics on the apparent horizon in two different approaches. These approaches depend on the definition of the components of the energy momentum tensor of dark energy and lead to the nonequilibrium and equilibrium descriptions of thermodynamics. We have seen that an entropy production term appears in the nonequilibrium description due to the violation of the standard continuity equation because of the definition of the dark energy components. Consequently, the usual first law of thermodynamics is violated in the nonequilibrium picture. We also have examined the second law of thermodynamics which illustrates that the total entropy evolution with time including the horizon entropy, the nonequilibrium entropy production term, and the entropy of all matter field and energy components is a nondecreasing function of time. We have found that the second law of thermodynamics is satisfied in both phantom and quintessence phases of the universe evolution. It should be noted that the validity of the second law of thermodynamics in the nonequilibrium picture imposes a constraint on the coupling parameter ; that is, an upper bound on can be deduced, which depends explicitly on the choices of functions and .

To have an equilibrium description of thermodynamics, we have redefined the dark energy density and pressure in a manner that the new components satisfy the continuity equation, so there are no extra entropy terms and the first law of thermodynamics holds. We have also shown that the change of the horizon entropy in the equilibrium picture involves the information of the change of the horizon entropy as well as the change of the entropy production term in the nonequilibrium description.

In studying the second law of thermodynamics, we have examined the evolution of the entropy contributed by the horizon entropy and all matter fields and energy contents in the equilibrium description. We have found the condition to satisfy the second law of thermodynamics which holds in the phantom phase of the universe. Furthermore, for the validity of the second law in the nonphantom phase it is required that and there is no constraint to the coupling parameter of the nonminimal gravity scenario.

Competing Interests

The authors declare that there are no competing interests regarding the publication of this paper.

Acknowledgments

The authors would like to acknowledge Professor Kourosh Nozari for invaluable remarks.

References

A. Einstein, “Riemannian geometry, while maintaining the notion of teleparallelism,” Sitzungsberichte der Preussischen Akademie der Wissenschaften, vol. 17, pp. 217–221, 1928.
View at: Google Scholar
K. Hayashi and T. Shirafuji, “New general relativity,” Physical Review D, vol. 19, no. 12, pp. 3524–3553, 1979.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
S. Capozziello, V. F. Cardone, S. Carloni, and A. Troisi, “Curvature quintessence matched with observational data,” International Journal of Modern Physics D, vol. 12, no. 10, pp. 1969–1982, 2003.
View at: Publisher Site | Google Scholar
S. Nojiri and S. D. Odintsov, “Modified f(R) gravity consistent with realistic cosmology: from a matter dominated epoch to a dark energy universe,” Physical Review D, vol. 74, no. 8, Article ID 086005, 2006.
View at: Publisher Site | Google Scholar
S. Nojiri and S. D. Odinstsov, “Introduction to modified gravity and gravitational alternative for dark energy,” International Journal of Geometric Methods in Modern Physics, vol. 4, no. 1, p. 115, 2007.
View at: Publisher Site | Google Scholar
T. P. Sotiriou and V. Faraoni, “f(R) theories of gravity,” Reviews of Modern Physics, vol. 82, no. 1, pp. 451–497, 2010.
View at: Publisher Site | Google Scholar | MathSciNet
K. Nozari and T. Azizi, “Phantom-like behavior in f(R)-gravity,” Physics Letters B, vol. 680, no. 3, pp. 205–211, 2009.
View at: Publisher Site | Google Scholar
R. Ferraro and F. Fiorini, “Modified teleparallel gravity: inflation without an inflaton,” Physical Review D, vol. 75, no. 8, Article ID 084031, 2007.
View at: Publisher Site | Google Scholar | MathSciNet
G. R. Bengochea and R. Ferraro, “Dark torsion as the cosmic speed-up,” Physical Review D, vol. 79, no. 12, Article ID 124019, 5 pages, 2009.
View at: Publisher Site | Google Scholar
E. V. Linder, “Einstein's other gravity and the acceleration of the Universe,” Physical Review D, vol. 81, Article ID 127301, 2010.
View at: Publisher Site | Google Scholar
M. H. Daouda, M. E. Rodrigues, and M. J. S. Houndjo, “New static solutions in f(T) theory,” European Physical Journal C, vol. 71, article 1817, 2011.
View at: Publisher Site | Google Scholar
K. Bamba, C. Geng, C. Lee, and L. Luo, “Equation of state for dark energy in f(T) gravity,” Journal of Cosmology and Astroparticle Physics, vol. 2011, article no. 021, 2011.
View at: Publisher Site | Google Scholar
J. B. Dent, S. Dutta, and E. N. Saridakis, “f(T) gravity mimicking dynamical dark energy. Background and perturbation analysis,” Journal of Cosmology and Astroparticle Physics, vol. 2011, 2011.
View at: Publisher Site | Google Scholar
Y. Cai, S. Capozziello, M. D. Laurentis, and E. N. Saridakis, “f(T) teleparallel gravity and cosmology,” Reports on Progress in Physics, vol. 79, no. 10, Article ID 106901, 2016.
View at: Publisher Site | Google Scholar
T. Harko, F. S. N. Lobo, G. Otalora, and E. N. Saridakis, “Nonminimal torsion-matter coupling extension of f(T) gravity,” Physical Review D, vol. 89, no. 12, Article ID 124036, 2014.
View at: Publisher Site | Google Scholar
C. J. Feng, F. Ge, X. Li, R. Lin, and X. Zhai, “Towards realistic f(T) models with nonminimal torsion-matter coupling extension,” Physical Review D, vol. 92, no. 10, Article ID 104038, 2015.
View at: Publisher Site | Google Scholar
S. Carloni, F. S. Lobo, G. Otalora, and E. N. Saridakis, “Dynamical system analysis for a nonminimal torsion-matter coupled gravity,” Physical Review D, vol. 93, no. 2, Article ID 024034, 2016.
View at: Google Scholar | MathSciNet
S. W. Hawking, “Particle creation by black holes,” Communications in Mathematical Physics, vol. 43, no. 3, pp. 199–220, 1975.
View at: Publisher Site | Google Scholar | MathSciNet
J. D. Bekenstein, “Black holes and entropy,” Physical Review. D, vol. 7, pp. 2333–2346, 1973.
View at: Publisher Site | Google Scholar | MathSciNet
T. Jacobson, “Thermodynamics of spacetime: the Einstein equation of state,” Physical Review Letters, vol. 75, no. 7, pp. 1260–1263, 1995.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
R.-G. Cai and S. P. Kim, “First law of thermodynamics and Friedmann equations of Friedmann-Robertson-Walker universe,” Journal of High Energy Physics, vol. 2005, 2005.
View at: Publisher Site | Google Scholar | MathSciNet
M. Akbar and R. Cai, “Thermodynamic behavior of the Friedmann equation at the apparent horizon of the FRW universe,” Physical Review D, vol. 75, no. 8, Article ID 084003, 2007.
View at: Publisher Site | Google Scholar
R.-G. Cai and L.-M. Cao, “Unified first law and the thermodynamics of the apparent horizon in the FRW universe,” Physical Review D, vol. 75, no. 6, Article ID 064008, 2007.
View at: Publisher Site | Google Scholar | MathSciNet
R.-G. Cai, L.-M. Cao, Y.-P. Hu, and S. Kim, “Generalized Vaidya spacetime in Lovelock gravity and thermodynamics on the apparent horizon,” Physical Review D, vol. 78, no. 12, Article ID 124012, 2008.
View at: Publisher Site | Google Scholar | MathSciNet
A. Sheykhi, B. Wang, and R.-G. Cai, “Deep connection between thermodynamics and gravity in Gauss-Bonnet braneworlds,” Physical Review D, vol. 76, no. 2, Article ID 023515, 5 pages, 2007.
View at: Publisher Site | Google Scholar | MathSciNet
S.-F. Wu, B. Wang, G.-H. Yang, and P.-M. Zhang, “The generalized second law of thermodynamics in generalized gravity theories,” Classical and Quantum Gravity, vol. 25, no. 23, Article ID 235018, 2008.
View at: Publisher Site | Google Scholar | MathSciNet
C. Eling, R. Guedens, and T. Jacobson, “Nonequilibrium thermodynamics of spacetime,” Physical Review Letters, vol. 96, no. 12, Article ID 121301, 2006.
View at: Publisher Site | Google Scholar | MathSciNet
K. Bamba and C.-Q. Geng, “Thermodynamics in f(R) gravity in the Palatini formalism,” Journal of Cosmology and Astroparticle Physics, vol. 2010, no. 6, article 014, 20 pages, 2010.
View at: Publisher Site | Google Scholar | MathSciNet
K. Bamba, C.-Q. Geng, and S. Tsujikawa, “Equilibrium thermodynamics in modified gravitational theories,” Physics Letters B, vol. 688, no. 1, pp. 101–109, 2010.
View at: Publisher Site | Google Scholar
K. Bamba and C. Geng, “Thermodynamics of cosmological horizons in f(T) gravity,” Journal of Cosmology and Astroparticle Physics, vol. 2011, article no. 008, 2011.
View at: Publisher Site | Google Scholar
K. Karami, M. S. Khaledian, and N. Abdollahi, “The generalized second law of gravitational thermodynamics on the apparent horizon in f(R)-gravity,” EPL, vol. 98, no. 3, 2012.
View at: Publisher Site | Google Scholar
K. Bamba, C. Geng, S. Nojiri, and S. D. Odintsov, “Equivalence of the modified gravity equation to the Clausius relation,” EPL (Europhysics Letters), vol. 89, no. 5, p. 50003, 2010.
View at: Publisher Site | Google Scholar
M. Sharif and M. Zubair, “Thermodynamics in f(R,T) theory of gravity,” Journal of Cosmology and Astroparticle Physics, vol. 3, article 28, 2012.
View at: Publisher Site | Google Scholar
T. Harko, “Thermodynamic interpretation of the generalized gravity models with geometry-matter coupling,” Physical Review D, vol. 90, no. 4, Article ID 044067, 2014.
View at: Publisher Site | Google Scholar
M. Sharif and M. Zubair, “Thermodynamics in f(R,T) theory of gravity,” Journal of Cosmology and Astroparticle Physics, vol. 2012, no. 3, article 028, 2012.
View at: Google Scholar
M. Sharif and M. Zubair, “Thermodynamics in modified gravity with curvature matter coupling,” Advances in High Energy Physics, vol. 2013, Article ID 947898, 10 pages, 2013.
View at: Publisher Site | Google Scholar | MathSciNet
T. Azizi and N. Borhani, “Thermodynamics in hybrid metric-Palatini gravity,” Astrophysics and Space Science, vol. 357, article 146, 2015.
View at: Publisher Site | Google Scholar
R. Miao, M. Li, and Y. Miao, “Violation of the first law of black hole thermodynamics in f(T) gravity,” Journal of Cosmology and Astroparticle Physics, vol. 2011, 2011.
View at: Publisher Site | Google Scholar
K. Karami and A. Abdolmaleki, “Generalized second law of thermodynamics in f(T) gravity,” Journal of Cosmology and Astroparticle Physics, vol. 2012, no. 4, article 007, 2012.
View at: Publisher Site | Google Scholar
K. Bamba and C. Geng, “Thermodynamics of cosmological horizons in f(T) gravity,” Journal of Cosmology and Astroparticle Physics, vol. 2011, 2011.
View at: Publisher Site | Google Scholar
M. Sharif and S. Rani, “Thermodynamics in f (T) gravity and corrected entropies,” The European Physical Journal Plus, vol. 128, article 96, 2013.
View at: Publisher Site | Google Scholar
M. Zubair and S. Waheed, “Energy conditions in f(T) gravity with non-minimal torsion-matter coupling,” Astrophysics and Space Science, vol. 355, no. 2, pp. 361–369, 2015.
View at: Publisher Site | Google Scholar
M. Zubair and S. Waheed, “Thermodynamic study in modified f(T) gravity with cosmological constant regime,” Astrophysics and Space Science, vol. 360, article 68, 2015.
View at: Publisher Site | Google Scholar
M. Zubair and A. Jawad, “Generalized second law of thermodynamics in f(T,T_G) gravity,” Astrophysics and Space Science, vol. 360, no. 1, article 11, 2015.
View at: Publisher Site | Google Scholar
M. Askin, H. Abedi, and M. Salti, “Thermodynamics in f(T,θ) gravity,” Romanian Journal of Physics, vol. 60, no. 1-2, pp. 4–55, 2015.
View at: Google Scholar
P. C. Davies, “Cosmological horizons and the generalised second law of thermodynamics,” Classical and Quantum Gravity, vol. 4, no. 6, pp. L225–L228, 1987.
View at: Publisher Site | Google Scholar
H. M. Sadjadi, “Generalized second law in a phantom-dominated universe,” Physical Review D, vol. 73, no. 6, 2006.
View at: Publisher Site | Google Scholar
M. R. Setare and S. Shafei, “A holographic model of dark energy and the thermodynamics of a non-flat accelerated expanding universe,” Journal of Cosmology and Astroparticle Physics, vol. 2006, article no. 011, 2006.
View at: Publisher Site | Google Scholar
E. Babichev, V. Dokuchaev, and Y. Eroshenko, “Black hole mass decreasing due to phantom energy accretion,” Physical Review Letters, vol. 93, no. 2, Article ID 021102, 2004.
View at: Publisher Site | Google Scholar
J. Zhou, B. Wang, Y. Gong, and E. Abdalla, “The generalized second law of thermodynamics in the accelerating universe,” Physics Letters. B, vol. 652, no. 2-3, pp. 86–91, 2007.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
G. Izquierdo and D. Pavón, “The generalized second law in phantom dominated universes in the presence of black holes,” Physics Letters B, vol. 639, no. 1, pp. 1–4, 2006.
View at: Publisher Site | Google Scholar
M. Jamil and M. Akbar, “Generalized second law of thermodynamics for a phantom energy accreting BTZ black hole,” General Relativity and Gravitation, vol. 43, no. 4, pp. 1061–1068, 2011.
View at: Publisher Site | Google Scholar | MathSciNet
A. Abdolmaleki, T. Najafi, and K. Karami, “Generalized second law of thermodynamics in scalar-tensor gravity,” Physical Review D, vol. 89, no. 10, Article ID 104041, 2014.
View at: Publisher Site | Google Scholar
S. Wu, B. Wang, G. Yang, and P. Zhang, “The generalized second law of thermodynamics in generalized gravity theories,” Classical and Quantum Gravity, vol. 25, no. 23, 2008.
View at: Publisher Site | Google Scholar
J. M. Bardeen, B. Carter, and S. W. Hawking, “The four laws of black hole mechanics,” Communications in Mathematical Physics, vol. 31, pp. 161–170, 1973.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
R. M. Wald, “Black hole entropy is the Noether charge,” Physical Review D, vol. 48, no. 8, pp. R3427–R3431, 1993.
View at: Publisher Site | Google Scholar | MathSciNet
R. Brustein and M. Hadad, “Einstein equations for generalized theories of gravity and the thermodynamic relation $δ Q = T δ S$ are equivalent,” Physical Review Letters, vol. 103, no. 10, Article ID 101301, 4 pages, 2009.
View at: Publisher Site | Google Scholar | MathSciNet
B. Li, T. P. Sotiriou, and J. D. Barrow, “Large-scale structure in f(T) gravity,” Physical Review D, vol. 83, no. 10, 2011.
View at: Publisher Site | Google Scholar
S. A. Hayward, “Unified first law of black-hole dynamics and relativistic thermodynamics,” Classical and Quantum Gravity, vol. 15, no. 10, pp. 3147–3162, 1998.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet

Copyright

Copyright © 2017 Tahereh Azizi and Najibeh Borhani. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP³.

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