# Generalized second law of thermodynamic in modified teleparallel theory

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## Abstract

This study is conducted to examine the validity of the generalized second law of thermodynamics (GSLT) in flat FRW for modified teleparallel gravity involving coupling between a scalar field with the torsion scalar *T* and the boundary term \(B=2\nabla _{\mu }T^{\mu }\). This theory is very useful, since it can reproduce other important well-known scalar field theories in suitable limits. The validity of the first and second law of thermodynamics at the apparent horizon is discussed for any coupling. As examples, we have also explored the validity of those thermodynamics laws in some new cosmological solutions under the theory. Additionally, we have also considered the logarithmic entropy corrected relation and discuss the GSLT at the apparent horizon.

## 1 Introduction

The rapid growth of observational measurements on expansion history reveals the expanding paradigm of the universe. This fact is based on accumulative observational evidence mainly from Type Ia supernova and other renowned sources [1, 2, 3]. The expanding phase implicates the presence of a repulsive force which compensates the attractiveness property of gravity on cosmological scales. This phenomenon may be translated as the existence of exotic matter components and most acceptable understanding for such enigma is termed dark energy (DE); it has a large negative pressure. Various DE models and modified theories of gravity have been proposed to incorporate the role of DE in cosmic expansion history (for reviews see [4, 5, 6]).

In contrast to Einstein’s relativity and its proposed modifications where the source of gravity is determined by curvature scalar terms, another formulation has been presented which comprises a torsional formulation as gravity source [7, 8, 9, 10, 11]. This theory is labeled TEGR (teleparallel equivalent of general relativity) and it is determined by a Lagrangian density involving a zero curvature Weitzenböck connection instead of a zero torsion Levi-Civita connection with the vierbein as a fundamental tool. The Weitzenböck connection is a specific connection which characterizes a globally flat space-time endowed with a non-zero torsion tensor. Using that connection, one can construct an alternative and equivalent theory of GR. The latter appears, since the scalar torsion only differs by a boundary term \(B=\frac{2}{e}\partial _\mu (eT^\mu )\) with the scalar curvature by the relationship \(R=-T+B\), making the two variations of the Einstein–Hilbert and TEGR actions the same. Thus, these two theories have the same field equations. However, these two theories have different geometrical interpretations, since in TEGR the torsion acts as a force; meanwhile in GR, the gravitational effects are understood due to the curved space-time. TEGR is then further extended to a generalized form by the inclusion of a *f*(*T*) function in the Lagrangian density (as *f*(*R*) is the extension of GR) and it has been tested cosmologically by numerous researchers [12, 13, 14]. It is important to mention that *f*(*T*) and *f*(*R*) are no longer equivalent theories, and in order to consider the equivalent teleparallel theory of *f*(*R*), one needs to consider a more generalized function *f*(*T*, *B*), incorporating the boundary term in the action [15]. In [16], the authors studied some cosmological features (reconstruction method, thermodynamics and stability) within *f*(*T*, *B*) gravity and, in [17], some cosmological solutions were found using the Noether symmetry approach. Additionally, it has been proved that when one considers Gauss–Bonnet higher order terms, an additional boundary term \(B_{G}\) (related to the contorsion tensor) needs to be introduced to find the equivalent teleparallel modified Gauss–Bonnet *f*(*R*, *G*) theory [18].

Later, Harko et al. [19] proposed a comprehensive form of this theory by involving a non-minimal torsion matter interaction in the Lagrangian density. In Ref. [20], Zubair and Waheed recently have investigated the validity of energy constraints for some specific *f*(*T*) models and discussed the feasible bounds of involved arbitrary parameters. They also discussed the validity of the generalized second law of thermodynamic in the cosmological constant regime [21].

Another very much studied approach in modified theories of gravity is to change the matter content of the universe by adding a scalar field in the matter sector. These models have been considered several times in cosmology, using different kinds of scalar fields such as quintessence, quintom, k-essence, etc. (see [22, 23, 24]). Moreover, we can also extend that idea by adding a coupling between the scalar field and the gravitational sector (see [25, 26, 27]) where, cosmologically speaking, we can have new interesting results such as the possibility of crossing the phantom barrier. Motivated by these theories, other interesting modified theories of gravity have also been discussed in the literature [28, 29, 30] on the cosmological landscape. Recently, Bahamonde and Wright [31] presented a new model of teleparallel gravity by introducing a scalar field non-minimally coupled to both the torsion *T* and the boundary term \(B=2\nabla _{\mu }T^{\mu }\). It is shown that such a theory can describe the non-minimal coupling to torsion and also the non-minimal coupling to scalar curvature under certain limits.

Black hole thermodynamics suggests that there is a fundamental connection between gravitation and thermodynamics [32]. Hawking radiation [33], the proportionality relation between the temperature and surface gravity, and also the connection between horizon entropy and the area of a black hole [34] support this idea. Jacobson [35] was the first to deduce the Einstein field equations from the Clausius relation, \(T_hd\hat{S}_h={\delta }\hat{Q}\), together with the fact that the entropy is proportional to the horizon area. In the case of a general spherically symmetric space-time, it was shown that the field equations can be constituted as the first law of thermodynamics (FLT) [36]. The relation between the FRW equations and the FLT was shown in [37] for \(T_h=1/2{\pi }R_A,~ S_h=\pi R_A^2/G\). The investigation of the validity of thermodynamical laws in GR as well as modified theories has been carried out by numerous researchers in the literature [38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53].

In this work we focus on the validity of the thermodynamical laws in a modified teleparallel gravity involving a non-minimal coupling between both torsion scalar and the boundary term with a scalar field.

In Sect. 1, we give a brief introduction of this theory and then we derive the respective field equation for flat FRW geometry with a perfect fluid as the matter contents. In Sects. 2 and 3, we formulate the first and second law of thermodynamics at the apparent horizon for any coupling. Additionally, as examples, we study the validity of the thermodynamics laws using new cosmological solutions based on power-law and exponential-law cosmology (see Appendix A). In Sect. 4, we discuss the validity of GSLT for the entropy functional with quantum corrections. In each case, suitable limits of the parameters are chosen in order to visualize the validity of GSLT in quintessence, scalar field non-minimally coupled to torsion (known as teleparallel dark energy) and non-minimally coupled to the scalar curvature theories. Finally, in Sect. 5, a discussion of the work is presented.

In the following paper, the notation used is the same as in [31], where the tetrad and the inverse of the tetrad fields are denoted by a lower letter \(e^{a}_{\mu }\) and a capital letter \(E^{\mu }_{a}\), respectively, with the \((+,-,-,-)\) metric signature.

## 2 Teleparallel quintessence with a non-minimal coupling to a boundary term

*T*and a boundary term defined in terms of a divergence of the torsion vector, \(B=\frac{2}{e}\partial _\mu (eT^\mu )\), where \(e=\text {det}(e^{a}_{\mu })\). The action of this theory is given by

*B*(NMC-B) is recovered if \(\xi =0\). Using dynamical system techniques, the cosmology in NMC-B was studied in [31]. If \(\chi =0\), we get the same action as in [25], which is known as “teleparallel dark-energy theory” (TDE). By choosing \(\chi =-\xi \) we obtain scalar field models non-minimally coupled to the Ricci scalar that hereafter, for simplicity we will label as NMC-R [26]. In addition, the so-called “Minimally coupled quintessence” theories arise when we take \(\chi =\xi =0\) [27].

*a*(

*t*) represents the scale factor and the corresponding tetrad components are \(e^i_\mu =(1,a(t),a(t),a(t))\). It is important to mention that despite

*f*(

*T*) gravity not being invariant under local Lorentz transformations, the latter tetrad is a “good tetrad” to consider, since it does not constrain its field equations [54]. Hence, this tetrad can be used safely within this scalar field theory too.

*w*being the state parameter. If we use the above equation, we can directly find that the energy density becomes

## 3 Thermodynamics in modified teleparallel theory

In this section we are interested to explore the general thermodynamic laws in the framework of the theory studied in [31], where the authors considered a quintessence theory non-minimally coupled between both a torsion scalar *T* and the boundary term *B* with the scalar field (NMC-B+T). The main aim of the next sections will be to formulate the first and second laws of thermodynamics in this theory. The complete and general thermodynamics law will be derived. After that, we will use the cosmological solutions found in Appendix A to study some interesting examples to visualize if they do or do not satisfy the thermodynamic laws.

### 3.1 First law of thermodynamics

This section is devoted to an investigation of the validity of the first law of thermodynamics in NMC-(B+T) at the apparent horizon for a flat FRW universe.

Initially Akbar and Cai used this approach to discuss thermodynamic laws in *f*(*R*) gravity [59]. It is shown that an additional entropy term is produced, to be compared to other modified theories. Later Bamba et al. [41, 42, 60, 61] developed the first law of thermodynamics in Palatini *f*(*R*), *f*(*T*), \(f(R,\phi ,X)\) (where \(X=-1/2g^{\mu \nu }\nabla _\mu \phi \nabla _\nu \phi \) is the kinetic term of a scalar field \(\phi \)) and \(f(R,\phi ,X,G)\) (where \(G=R^2-4R_{\mu \nu }R^{\mu \nu }+R_{\mu \nu \rho \sigma }R^{\mu \nu \rho \sigma }\) is the Gauss–Bonnet invariant) theories and formulated an additional entropy production term. A similar approach is applied to discuss the thermodynamic laws in *f*(*R*, *T*), \(f(R,L_m)\) and \(f(R,T,R_{\mu \nu }T^{\mu \nu })\) theories, and one can see that the presence of non-equilibrium entropy production terms is necessary in such theories [46, 47, 62]. Bamba et al. [41, 42, 60, 61] have shown that one can manipulate the FRW equations in order to redefine the entropy relation, which results in an equilibrium description of thermodynamics so that the first law of thermodynamics takes the form \(T\mathrm{d}S=\mathrm{d}E-W\mathrm{d}V\). Moreover, in all these theories it has been the case that the usual form of the first law of thermodynamics, i.e., \(T\mathrm{d}S_\mathrm{eff}=\mathrm{d}E-W\mathrm{d}V\), can be obtained by defining the general entropy relation as a sum of an horizon entropy and an entropy production term.

*T*and a boundary term at the apparent horizon of FRW space-time.

### 3.2 Generalized second law of thermodynamics

*i*th component. Summing up the total entropy inside the horizon, we find

#### 3.2.1 Specific model: power-law solutions for \(f(\phi )=1+\xi \kappa ^2 \phi ^2\) and \(g(\phi )=\chi \kappa ^2\phi ^2\)

In this section we will analyze if the GSLT is valid for different power-law models. In Sect. A.1, we found some specific solutions to the specific coupling \(f(\phi )=1+\xi \kappa ^2 \phi ^2\) and \(g(\phi )=\chi \kappa ^2 \phi ^2\). Here, we will focus our study for quartic (\(m=-1\)) and inverse potentials (\(m=2/3\)). Additionally, by setting the coupling constants \(\xi \) and \(\chi \), we will also focus our study on different non-minimally coupled scalar–tensor theories.

**(a) Power-law potential with** \(\chi \ne \frac{1}{4}\)

Here, we set the validity condition for GSLT in terms of the parameters *m*, \(\xi \) and \(\chi \). We choose *m* to show the particular representation of potential.

**(b) Case** \(m=-1\)

*m*corresponds to a quartic potential, \(V(\phi )\propto \phi ^4\). Here, we find the relation for

*n*of the form \(n=\frac{1-6\chi }{6\xi +\chi }\) and the constraint \(n>1\) results in the following condition:

**(c) Case** \(m=2/3\)

**(d) Power-law potential with** \(\chi =\frac{1}{4}\)

*n*, and validity is shown in Fig. 6.

#### 3.2.2 Specific model: exponential solutions for \(f(\phi )=1+\kappa ^2\alpha e^{m\phi }\) and \(g(\phi )=\kappa ^2\beta e^{k\phi }\)

## 4 Generalized second law of thermodynamics with logarithmic entropy corrections

## 5 Conclusions

The non-minimally coupling models in cosmology are frequently used to study guaranteed late time acceleration, phantom crossing and the existence of finite time future singularity. In this paper, the thermodynamic study is executed in modified teleparallel theory which involves scalar field non-minimally coupled to both the torsion and a boundary term [31]. One interesting factor in this theory is that in a suitable limit, one can recover very well-known theories of gravity such as quintessence, teleparallel dark energy and non-minimally coupled scalar field with the Ricci scalar *R*.

In this paper, we investigated the GSLT for an expanding universe with apparent horizon. The laws of thermodynamics are universally valid and hence they must apply in a modified form to the whole universe. The existence of an apparent horizon in a space-time provides the opportunity to formulate the first law and the generalized second law of thermodynamics. According to the laws of thermodynamics, the first law must always be true for a physical system (provided it is non-dissipative); however, the GSLT holds exactly using the apparent horizon. We obtained these laws in the non-minimal coupled teleparallel theory with a boundary term and a scalar field. Our results generalize many previous GSLT studies, while extending them to the logarithmic corrected entropy–area law. Moreover, GSLT with quantum corrections have natural implications in the studies of the very early universe, since quantum corrections are directly linked with high energy and short distance scales.

We explored the existence of power-law solutions for two different choices of Lagrangian coefficients, which are functions of the scalar field, \(f(\phi )\) and \(g(\phi )\). In the first case we set \(f(\phi )=1+\kappa ^2\xi \phi ^2\) and \(g(\phi )=\kappa ^2\chi \phi ^2\) and choose the power-law forms of scale factor and scalar field to discuss possible forms of the power-law potential \(V(\phi )\) (see [28]). Here, one can recover the quadratic, quartic, inverse and Ratra–Peebles potentials. We also discuss the Brans–Dicke theory as a particular case in power-law cosmology. Moreover, we also discuss the power-law solutions for the choice of \(f(\phi )=1+\kappa ^2\alpha e^{m\phi }\) and \(g(\phi )=\kappa ^2\beta e^{k\phi }\) with \(\phi (t)=\frac{\phi _0}{m}\log (t)\). Here, we explored different forms of \(V(\phi )\) depending on the equation of state parameter.

We showed the behavior of matter density \(\rho \) and potential \(V(\phi )\) for the power-law potential; it can be seen that both are decreasing functions of time as shown in Fig. 1. In our discussion of the validity of GSLT, we considered the quartic and inverse potentials in all the viable special cases like TDE, NMC-B, NMC-R and NMC-(B+R) and presented results in Figs. 2, 3, 4, 5 and 6. In Figs. 2 and 3, we found that GSLT is true for NMC-B (with \(0<\xi <\frac{1}{6}\)) and TDE (with \(0<\chi <\frac{1}{12}\)). For the inverse potential, we find that GSLT is valid for NMC-B (with \(\frac{-1}{8}<\chi <0\)) and TDE (with \(-\frac{2}{3}<\xi <0\)), whereas it is violated for TDE, as shown in Figs. 4 and 5. For the choice of \(\chi =\frac{1}{4}\), we found the validity region only in the case of NMC-(B+R) with \(n>\frac{1}{2}\) as shown in Fig. 6. For the exponential forms of the Lagrangian coefficients, we found that GSLT is always true for (\(\phi _0\le -2\) & \(\alpha <0\)) and (\(\phi _0\ge -2\) & \(\alpha >0\)), whereas for other choices of the parameters the validity of GSLT is time dependent. Finally, we also formulated the GSLT for logarithmic entropy corrections.

## Notes

### Acknowledgements

S.B. is supported by the Comisión Nacional de Investigación Científica y Tecnológica (Becas Chile Grant No. 72150066). The authors would like to thank the anonymous referee for the interesting feedback and useful comments on the paper.

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