Advances in High Energy Physics

Volume 2016 (2016), Article ID 7020162, 12 pages

http://dx.doi.org/10.1155/2016/7020162

## Isotropic Stars in Higher-Order Torsion Scalar Theories

^{1}Centre for Theoretical Physics, The British University in Egypt, P.O. Box 43, El-Sherouk City 11837, Egypt^{2}Mathematics Department, Faculty of Science, Ain Shams University, Cairo 11566, Egypt

Received 20 September 2015; Revised 1 January 2016; Accepted 5 January 2016

Academic Editor: Chao-Qiang Geng

Copyright © 2016 Gamal G. L. Nashed. 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 SCOAP^{3}.

#### Abstract

Two different nondiagonal tetrad spaces reproducing spherically symmetric spacetime are applied to the field equations of higher-order torsion scalar theories. Assuming the existence of conformal Killing vector, two isotropic solutions are derived. We show that the first solution is not stable while the second one confirms a stable behavior. We also discuss the construction of the stellar model and show that one of our solutions is capable of such construction while the other is not. Finally, we discuss the generalized Tolman-Oppenheimer-Volkoff and show that one of our models has a tendency to equilibrium.

#### 1. Introduction

It is well known that in gravity not only inflation [1] in the early universe but also late time cosmic acceleration can be realized [2–27]. Recently, there are many models constructed to describe dark energy without the use of cosmological constant (for more details, see review [28] and references therein). The main merit of gravity is that its gravitational field equation is second order as GR. There are arguments in terms of theoretical properties of gravity, for example, local Lorentz invariance [29–31], nonminimal coupling of teleparallel gravity to a scalar field [32–34], and nonlinear causality [35]. Recently, number of gravitational theories have been proposed [36–62]. The structures of neutron and quark stars in theory of gravity have been investigated [63]. The anisotropic behavior, regularity conditions, stability, and surface redshift of the compact stars have been checked [64]. Under those theories, it is shown that are not dynamically identical to teleparallel action plus a scalar field [61]. It has been shown that investigations of , using observational data, are compatible with observations (see, e.g., [65, 66] and references therein). A new type of theory was proposed in order to explain the acceleration phase of the universe [60]. Also, it has been shown that the well-known problem of frame dependence and violation of local Lorentz invariance in the formulation of gravity is a consequence of neglecting the role of spin connection [31].

theory coupled with anisotropic fluid has been examined for static spacetimes with spherical symmetry and many classes of solutions have been derived [67]. It has been shown that some conditions on the coordinates, energy density, and pressures can produce new classes of anisotropic and isotropic solutions. Some of new black holes and wormholes solutions have been derived by selecting a set of nondiagonal tetrads [68]. It has been shown that relativistic stars can exist in the frame of and static spherically symmetric perfect fluid solutions have been derived [69]. A special analytic vacuum spherically symmetric solution with constant torsion scalar, within the framework of , has been derived [70]. D-dimensional charged flat horizon solutions have been derived for a specific form of ; that is, [71]. A complete investigation of the Noether symmetry approach in gravity at FRW and spherical levels, respectively, has been investigated [72]. In the framework of gravitational theories, there are many solutions, spherically symmetric [59], spherically symmetric charged [73], homogenous anisotropic [74], and stability of the Einstein static closed and open universe [75]. Some cosmological features of the CDM model in the framework of the are investigated [76]. However, till now, no spherically symmetric isotropic solution,* using nondiagonal tetrad fields*, is derived in this theory. It is the aim of the present study to find an analytic, isotropic spherically symmetric solution in higher-order torsion scalar theories. The arrangement of this study is as follows: in Section 2, ingredients of gravitational theory are provided. In Section 3, two different tetrad spaces having spherical symmetry are applied to the field equations of . Assuming the conformal Killing vector (CKV), we derived two nonvacuum spherically symmetric solutions in Section 3. The physics relevant to the derived solutions are analyzed in Section 4. The energy conditions are satisfied for the two solutions provided that the constants of integration are positive. In addition, the stability condition, the nature of the star, and Tolman-Oppenheimer-Volkoff (TOV) equation are shown to be satisfied for one solution. The results obtained in this study are discussed in final section.

#### 2. Ingredients of Gravitational Theory

Another description of Einstein’s general relativity (GR) of gravitation is done through the employment of what is called teleparallel equivalent of general relativity (TEGR). The ingredient quantity of this theory is the vierbein (tetrad) fields (Greek letters indicate spacetime indices while Latin letters running from to describe Lorentz indices. Time and space indices are denoted as and , where ) alternative to metric tensor fields . The associated metric space can be constructed from the vierbein fields with being the Minkowskian metric of the tangent space; thus, the Levi-Civita symmetric connection is constructed from the metric and its first derivative [77]. Within TEGR, it is possible to build a nonsymmetric connection, Weitzenböck, . The tetrad 4-space is depicted as a pair , where is a -dimensional smooth manifold and () are -linearly independent vector fields defined globally on . The tetrad space has a main merit that is the vanishing of the vierbein’s covariant derivative; that is, , where the covariant derivative, , is regarding the nonsymmetric Weitzenböck connection. Therefore, the vanishing of the vierbein’s covariant derivative recognizes autoparallelism or absolute parallelism condition. Actually, the operator is not invariant under local Lorentz transformations (LLT). In this respect, the symmetric metric (10 degrees of freedom) cannot guess one set of vierbein fields; then, the extra degrees of freedom need to be determined so as one physical frame is used. The vector fields are called the parallelization vector fields. Because of the absolute parallelism condition, it can be shown that the metricity condition is satisfied. The Weitzenböck connection is curvatureless while it has a nonvanishing torsion tensor given asand contortion tensor asThe teleparallel torsion scalar which reproduces the TEGR theory is given bywhere the tensor of type is defined aswhich is skew symmetric in the last two indices. Similar to the theory, one can define the action of theory asand we have assumed the units in which and are the matter fields. Considering the action (5) as a function of the fields and putting the variation of the function regarding the field to be vanishing, one can obtain the following equations of motion [36, 71]:where , , , and denotes the energy-momentum tensor of the anisotropic fluid which is defined aswith representing the radial pressure, representing the tangential pressure, andEquations (6) are the field equations of gravitational theory.

#### 3. Nonvacuum Spherically Symmetric Solutions in Higher-Order Torsion Scalar Theories

In this section, we are going to apply two, nondiagonal, different tetrad fields having spherical symmetry to the field equations (6).

##### 3.1. First Tetrad

The equation of motion of GR supplies rich field to use symmetries which link geometry and matter in a natural way. Collineations are symmetries which come from either geometrical viewpoint or physical relevant quantities. The importance of collineations is the CKV which provides more information of the construction of the spacetime geometry. In the language of mathematics, CKV are the motions in which the metric tensor of a spacetime becomes invariant up to a scale factor. Also, the employment of the CKV simplifies the generation of exact solutions to the equations of motions of GR. This is done by reducing the complicated nonlinear partial differential equations of GR to simple ordinary differential equations. The CKV is defined aswith being the Lie derivative operator of the metric tensor and the conformal factor. One can assume the vector which creates the conformal symmetry and makes the metric conformally mapped onto itself through . One must note that and are not necessarily static even supposing a static metric [78, 79]. In addition, one must be careful about the following: (i)If , then (9) leads to a Killing vector.(ii)If constant, then (9) leads to homothetic vector.(iii)If , then (9) yields conformal vectors. Furthermore, if , then the spacetime becomes asymptotically flat and one has a null Weyl tensor. Thus, to have more understanding of the spacetime geometry, one must take into account the CKV. Essentially, the Lie derivative operator shows the interior gravitational field of a stellar configuration related to the vector field .

The first tetrad field having a stationary and spherical symmetry with local Lorentz transformations has the following form [80]:where and are two unknown functions of the radial coordinate, .

The associated metric of (10) takes the following form:which is a static spherically symmetric spacetime that admits one parameter group of conformal motion. Equation (11) is conformally mapped onto itself along . Therefore, (9) leads towhere 0 and 1 refer to the temporal and spatial coordinates and , respectively. The above set of equations lead towith , , , and being constants of integration.

Using (13), tetrad (10) is rewritten as

Tetrad field (14) has the following associated metric:Using (14) in (3), we get the scalar torsion in the following form:Using (16) and (14) in the field equations (6), we get the following nonvanishing components:Second equation of (17) leads to , or . The case gives a constant function and this is out of the scope of the present study. Therefore, we are searching for solutions that make constraint on the form of have the form:Assuming the isotropic conditionand using (19) in (17), we getThe sound velocity is defined as . Using (20), we get the sound velocity in the following form:

##### 3.2. Second Tetrad

The second tetrad space having a stationary and spherical symmetry takes the following form [73]:where and are two unknown functions of the radial coordinate, . Using the same procedure applied to tetrad (10), we get the following equations of CKV of tetrad (22):The above set of equations implywhere , , and are constants of integration.

Using (24), tetrad (22) can be rewritten asUsing (25), the torsion scalar (3) takes the following form:Inserting (26) and the components of the tensors and in the field equations (6), we obtainThe above system cannot be solved without assuming some specific constraint on the form of . Therefore, we are going to use the constraint (18) in (27) and obtain the following:Using (28), the sound velocity takes the following form:

#### 4. Physics Relevant to the Models

##### 4.1. Energy Conditions

Energy conditions are essential tools to understand various cosmological geometries and some general results related to the strong gravitational fields. These tools are three forms of energy conditions, the strong energy (SEC), null energy (NEC), and weak energy conditions (WEC) [81–83]. Such conditions have the following inequalities:

It is interesting to remember that the breaking of the conditions leads to the existence of the ghost instabilities.

##### 4.2. Energy Conditions of Smooth Transition Models

Let us apply the above procedure of the energy conditions given by (30) to the derived solutions given in the previous section. Equations (28) show that density has a positive value and , are satisfied with constant for the first model and for the second model as shown in Figures 1 and 2. This means that NEC, SEC, and WEC are satisfied for the above two models. Also, it is interesting to note that the density and pressure of both solutions do not depend on the constants and .