Generalized Second Law of Thermodynamics in Extended Theories of Gravity

By employing the general expression of temperature Th = |κ|/2π = 1 2πr̃A (1 − ̇̃ rA 2Hr̃A ) associated with the apparent horizon of a FRW universe and assuming a region of FRW universe enclosed by the apparent horizon as a thermal system in equilibrium, we are able to show that the generalized second law of thermodynamics holds in Gauss-Bonnet gravity and in more general Lovelock gravity. PACS numbers: 04.70.Dy, 97.60.Lf Email address: [email protected] Email address: [email protected] Introduction: It is possible to associate the notions of temperature and entropy with the apparent horizon of FRW universe analogous to the Hawking temperature and entropy associated with the black hole horizon [1, 2, 3]. The thermodynamic properties of the apparent horizon of FRW universe has been studied in various theories of gravity by many authors (see for examples [1, 2, 4, 6, 5, 7, 8]). The thermodynamic extension has also been made at the apparent horizon in the braneworld cosmology [9, 10, 11, 12]. In case of black holes horizon, the study of horizon thermodynamic has also been made by many authors (see for examples [22, 23, 24, 25]). More recently, Cai et al [5] has shown that by employing Clausius relation, δQ = ThdSh, to the apparent horizon of a FRW universe, they are able to reproduce the modified Friedmann equations by using quantum corrected entropy-area relation. Since the extra higher derivative terms in Gauss-Bonnet and Lovelock gravities can be thought of as quantum corrections to the Einstein gravity and Friemann equations at apparent horizon in these gravities can be cast as a first law of thermodynamics which leads to suggest that the thermodynamic interpretation of field equations at apparent horizon remains valid even if one includes the possible quantum corrections to the Einstein gravity. However, the thermodynamic interpretation of gravity as a first law at the apparent horizon of FRW universe in various gravity theories has been concentrated only for spherically symmetric spacetime metric. If thermodynamic interpretation of gravity near apparent horizon is to be generic, one needs to verify whether the results may hold not only for more general spacetimes but also for the other principles of thermodynamics, especially the generalized second law as global accepted principle in the universe. As a first step, we investigate further the thermodynamic of the apparent horizon of FRW universe in the case of generalized second law of thermodynamics. In this direction some work to suggest constraints on the parameters of the gravity theory has been made for the validity of second law [16]. A similar case for f(R) gravity with equilibrium thermodynamics has been considered in [18] and also see [13, 14, 15] and reference therein. By applying the general expression of temperature at apparent horizon of FRW universe, it has been shown [17] that the generalized second law holds in Einstein gravity. Hence, it is important to investigate whether the generalized law still holds within a region of FRW universe enclosed by the apparent horizon even if one includes the possible quantum corrections to the Einstein gravity where the entropy-area relation no longer holds. In this paper, We will discuss this issue in case of Gauss-Bonnet and more general lovelock gravities. Let us now start with a (n+1)-dimensional FRW universe of metric ds = −dt + a(t)( dr 2 1 − kr2 + r dΩn−1), (1) where dΩn−1 stands for the line element of (n-1)-dimensional unit sphere and the spatial curvature constant k = +1, 0 and −1 represents to a closed, flat and open universe, respectively. The above metric (1) can be rewritten in spherical form ds = habdx dx + r̃dΩn−1, (2)