Generalized second law of thermodynamic in modified teleparallel theory
Eur. Phys. J. C
Generalized second law of thermodynamic in modified teleparallel theory
M. Zubair 0
Sebastian Bahamonde 1
Mubasher Jamil 2
0 Department of Mathematics, COMSATS Institute of Information Technology Lahore , Lahore , Pakistan
1 Department of Mathematics, University College London , Gower Street, London WC1E 6BT , UK
2 Department of Mathematics, School of Natural Sciences (SNS), National University of Sciences and Technology (NUST) , H-12, Islamabad , Pakistan
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??T ?. 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?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?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?11
]. This theory is labeled TEGR (teleparallel equivalent
of general relativity) and it is determined by a Lagrangian
density involving a zero curvature Weitzenb?ck connection
instead of a zero torsion Levi-Civita connection with the
vierbein as a fundamental tool. The Weitzenb?ck 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
2
only differs by a boundary term B = e ??(eT ?) 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?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 BG (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?24]). Moreover, we can also
extend that idea by (...truncated)