Study of GSLT in Curvature-Matter Coupling Gravity

In this work, we study the rst and generalized second laws of thermodynamics at the apparent horizon of homogeneous and isotropic universe model in the context of f (G, T ) gravity (G and T represent the Gauss-Bonnet invariant and trace of the energymomentum tensor, respectively). We formulate the corresponding eld equations as well as determine the radius, temperature and entropy to analyze these laws. An extra term associated with entropy production is appeared in the rst law due to the non-equilibrium treatment of thermodynamics. It is found that the universal condition is obtained to preserve the generalized second law of thermodynamics.


Introduction
Thermodynamics has been a subject of great interest to explore the fascinating characteristics of matter variables in general relativity (GR) as well as in modied gravitational theories. For isotropic and homogeneous universe, Einstein eld equations can be expressed in terms of rst law of thermodynamics (FLT) [1]. Akbar and Cai [2] found that Friedmann equations evaluated at the apparent horizon can be rewritten in the form dE = τdS + WdV (E, τ, S , V and W are the energy, temperature, entropy, volume inside the horizon and work density, respectively) in the background of GR, Gauss-Bonnet (GB) gravity and the general Lovelock theory. It is found that an auxiliary entropy production term corresponding to the non-equilibrium treatment of thermodynamics is appeared in Clausius relation in modied theories of gravity while no such additional term is obtained in braneworld, GB and Lovelock gravitational theories [3]. Wu et al. [4] formulated the universal condition to check the validity of generalized second law of thermodynamics (GSLT) in the context of modied theories of gravity. Sadjadi [5] explored the validity of second law and GSLT for power-law solution as well as de Sitter universe in f (R, G) gravity. Abdolmaleki and Naja [6] investigated GSLT in f (G) gravity for isotropic and homogeneous universe lled with matter and radiation enclosed by apparent horizon with Hawking temperature.
In this paper, we explore the rst and GSLT at the apparent horizon in f (G, T ) gravity. This mod-ied gravitational theory deals with the non-minimal coupling between quadratic curvature invariant (a linear combination of Ricci scalar (R), Ricci (R αβ ) and Riemann (R αβξη ) tensors) and matter. The paper has the following format. In the next section, we construct the corresponding eld equations for isotropic and homogeneous universe with any spatial curvature while section 3 investigates the laws of thermodynamics at the apparent horizon of universe model. We summarize the results in the last section.

f (G, T ) Gravity
The action of f (G, T ) gravity is given by [7] where G = R αβξη R αβξη − 4R αβ R αβ + R 2 , g, L m and G denote determinant of the metric tensor (g αβ ), Lagrangian density associated with matter conguration and gravitational constant, respectively. The suitable form of generic function f (G, T ) describing the non-minimal coupling between curvature and matter is of the form The variation of the action (1) with respect to g αβ for the above model in the presence of pressureless uid yields the following eld equations where , is a covariant derivative) and prime represents derivative with respect to the corresponding variable. The line element for homogeneous and isotropic universe is where χ = a(t)r, a(t) and K represent the scale factor and spatial curvature parameter associated with open, at and closed cosmic geometries for K = −1, 0 and 1, respectively. Using Eqs. (3) and (4), we obtain where ρ (hc) and p (hc) are the higher order curvature terms given by a is a Hubble parameter. The subscripts G and T denote derivatives of Δ with respect to G and T , respectively whereas dot represents time derivative.
In this section, we discuss the rst as well as GSLT at the apparent horizon of homogeneous and isotropic universe in the background of f (G, T ) gravity.

First Law
Here, we construct the FLT which is based on the concept of energy conservation for model (2). The relation h αβ ∂ α χ∂ β χ = 0, where h αβ = diag(1, −a 2 (t) 1−Kr 2 ) is a two-dimensional line element, provides the radius of apparent horizon for the FRW universe model as To measure the innitesimal change in apparent horizon radius, we take the derivative of the above equation with respect to time and using Eq. (6), it follows that where dt represents the corresponding small time interval. The temperature on the apparent horizon is given by [1] where is the surface gravity. For homogeneous and isotropic universe model, we have Bekenstein-Hawking entropy is measured in units of Newton's gravitational constant dened as one fourth of area of apparent horizon (A = 4πχ 2 (ah) ) [8]. The entropy of stationary black hole solutions with bifurcate Killing horizons in the context of modied gravitational theories is a Noether charge entropy also dubbed as Wald entropy [9]. Brustein et al. [10] presented that this entropy is equal to quarter of apparent horizon area in units of effective gravitational coupling in these modied theories.
Wald entropy in f (G, T ) gravity is given by It is worth mentioning here that the entropy in f (G) gravity is obtained for f 3 (T ) = 0 while this formula corresponds to GR for Δ = 0 with B = 1 [11]. Using Eqs.(8)- (10), it follows that The total energy inside χ (ah) for homogeneous and isotropic universe model is Using Eqs. (11) and (12), we obtain where the innitesimal change dE is caused by the small displacement in horizon radius χ (ah) and measures the work done by the system. The FLT for model (2) can be expressed as where is the entropy production term which demonstrates the non-equilibrium behavior of thermodynamics. This shows that the eld equations for model (2) do not meet with the universal form of FLT (dE = τdS + WdV) due to the presence of this auxiliary term.

Generalized Second Law
The GSLT is associated with the total entropy of the system and is dened as where d i  S = ∂ t (d i S ) and S (tot) is associated with the entropy due to all matter contents present inside the horizon. The relationship between total entropy, energy density and pressure in a form of differential equation (Gibbs equation) is given by [4] τ (tot) dS (tot) = d(ρ (tot) V) + p (tot) dV, where τ (tot) measures the total temperature of all contents within the horizon and is related to the temperature at the apparent horizon as τ (tot) = λτ, 0 < λ < 1.