Anisotropic compact objects in f(T) gravity with Finch–Skea geometry
Abstract
We study relativistic solutions of anisotropic compact stars with Finch–Skea (FS) metric in f(T) gravity framework. The modified FS geometry is considered to obtain the equation of state (EoS) for different known stellar objects with given mass and radius. The modified Chaplygin gas (MCG) EoS is also considered to obtain stellar objects as the EoS inside the star is not yet known. The results obtained here is important in the two cases to understand properties of known stars, which are however not known observationally. The physical features of known stars are analyzed here and found that compact star formation may be possible with repulsive core. In the case of MCG in f(T) gravity compact stars may be obtained with anisotropic fluid (\(p_t>p_r\)), with maximum anisotropy at the center of the star, which however is not found when MCG is absent.
1 Introduction
It is known that General theory of Relativity (GR) is a geometric theory of gravitation based on the assumption that gravity manifests itself as the curvature of spacetime. In GR the spacetime structure may be determined using Lorentzian metric (g) and a linear LeviCivita connection (\(\Gamma \)). Although GR is a fairly successful theory it presents some serious issues at ultraviolet and infrared limits. Recent observational evidences of Galactic, extra Galactic and cosmic dynamics can not be understood by GR unless one considers some exotic forms of matter energy viz. dark matter and dark energy [1, 2]. Alternatively, one can modify the gravitational sector to fit the missing matterenergy of the observed universe. In recent times there is a huge effort to modify gravity [3, 4, 5, 6] to describe the evolution of the observed universe as well as to solve the issues of nonrenormalizability [7, 8] in GR.
The spacetime connections considered by Einstein are torsionless. However apart from simplicity there is no reason to consider that spacetime is torsionless. Torsion in the theory may arise from the consideration of spin or from a gradient of scalar field. The issues to modify GR with torsion [9, 10, 11] arises whether the spacetime connection is a symmetric or not. Essentially GR is a classical theory which does not accounted quantum effects, a situation which deals with gravity at a fundamental level. The paradigm is that the massenergy is the source of curvature and in general, spin is the source of torsion.
The consideration of a spacetime with torsion is an alternate approach equivalent to GR, was first introduced by Einstein himself known as Teleparallel Equivalent of General Relativity (TEGR) [12, 13, 14]. The most significant difference between GR and TEGR is that in the later case tetrad fields are present [15, 16]. In this case the tetrad fields are used to define a linear antisymmetric Weitzenböck connection related to torsion without curvature [17, 18]. Although both GR and TEGR provides similar results there are some fundamental differences between the two theories. According to GR, curvature represents the geometrical picture of spacetime which describes gravity. In contrast TEGR represents the same gravitational interactions in terms of torsion which acts as a force. This implies that in TEGR the geodesic equations are analogous to the Lorentz force equations of electrodynamics [19].
The need of constructing a modified gravitational theory arose from the observational evidences that our universe is passing through an accelerating phase of expansion [20]. Most of the works in the literature addressed curvaturebased formulation, and modify the Einstein–Hilbert action with f(R) paradigm in which the Lagrangian is considered to be a nonlinear function of the curvature scalar. However, one can reasonably think to start from TEGR, and use it as a basis to build a gravitational modification. The simplest class of these torsionbased modifications is the paradigm of f(T) gravity [21, 22, 23, 24], in which the Lagrangian is taken to be a nonlinear function of the TEGR Lagrangian T. TEGR coincides completely with general relativity at the level of equations, f(T) gravity may be simple compared to f(R) gravity as they give rise to second order differential equations whereas it is fourth order in case of f(R) theories.
In recent times, TEGR formalism and f(T) gravity are found successful to study cosmological and astrophysical phenomena. Ferraro and Fiorini [21] solved the particle horizon problem in a spatially flat FRW universe considering a modified version of TEGR in Born–Infeld approach. The solution of the equations of motion for the extended BTZ black hole with a cosmological constant has been studied in this model [22]. Li et al. [25] showed that in f(T) gravity the action and the field equations are not invariant under local Lorentz transformations, however, the usual teleparallel Lagrangian (TEGR) is an exception. The first law of blackhole thermodynamics is violated in f(T) framework which is a result of the lack of local Lorentz invariance [26]. Recently Wang et al. [27] studied spherically symmetric static solution in f(T) models with Maxwell terms in a particular frame based on the conformally Cartesian coordinates. It is found that only a limited class of f(T) models can be solved in this frame. Momeni et al. [28] obtained a new exact solution of anisotropic star in f(T) gravity considering Krori and Barua metric. Deliduman et al. [29] investigated neutron stars under modified f(T) gravity framework and found that, the relativistic neutron star solution in f(T) gravity models is possible only if f(T) is a linear function of the torsion scalar T, that is in the case of TEGR.
The motivation of the present paper is to obtain relativistic solution for an anisotropic compact object whose interior spacetime is described by the modified Finch–Skea (FS) geometry, for linear form of f(T) gravity. The metric proposed by Dourah and Ray [30] was modified by Finch and Skea [31] for investigating compact objects under various situations. Recently, FS geometry was modified to accommodate isotropic charged star [32] and it was also modified further incorporating anisotropic stars [33]. A number of research work in relativistic astrophysics using FS metric in 4 dimensions [34, 35, 36, 37, 38, 39, 40] and higher dimensions [41, 42] also came up.
In order to study the physical properties of a stellar system, it is important to predict a proper equation of state (EoS), i.e., \(p=p(\rho )\). In the absence of a reliable information about the EoS at very high densities, assumption of the metric potentials, based on the geometry has been found to be a reasonable approach to construct a stellar model. In our paper, we obtain EoS of a compact object with FS geometry in a linear f(T) gravity which corresponds to TEGR.
With the introduction of the existence of dark energy, Lobo [43] and coworkers considered stellar models which are called dark stars where the equation of state was taken in the linear form given by \(p=\omega \rho \) with \(1<\omega <1/3\). Chaplygin gas was considered to obtain acceleration of the universe and structure formation. Later a modification to the Chaplygin EoS with a more generalized form of equation of state [44, 45]. Dark energy provides sufficient repulsive force to counteract the continued gravitational collapse of the dark stars. The process reaches a stable configuration which is free from horizons and singularities. Recently Saha et al. [46] studied anisotropic stellar models with interior spacetime geometry described by KroriBarua metric with modified Chaplygin gas in f(T) gravity. Motivated by the above works, we also analyzed anisotropic compact stellar models with Finch–Skea geometry in f(T) framework with modified Chaplygin gas (MCG).
The paper is organized as follows: In Sect. 2, fundamentals of f(T) gravity is presented. The general form of the field equation is obtained by varying the action with respect to the tetrad field. In Sect. 3, the field equations are presented for a spherically symmetric static spacetime in terms of the torsion scalar, T and it’s derivative, and we have discussed that for a neutron star model only the linear form of f(T) (TEGR) is permitted. In Sect. 4, the exterior region of the stellar model is described by Schwarzschild metric and the junction conditions joining the interior modified FS metric and exterior region have been obtained. In Sect. 5, the field equations are solved and determine the unknown parameters from matching conditions and studied the various physical features of compact object e.g. energydensity, pressure, anisotropy etc. We analyzed the stability condition also and obtained a mass–radius relation. The probable EoS for the matter are determined. In Sect. 6, we compare the results obtained from the f(T) model to the results obtained previously in GR. In Sect. 7, for a compact object the EoS described by the modified Chaplygin gas in a f(T) gravity is considered to describe the physical features. Finally in Sect. 8, we summarize the results obtained.
2 Field equations of f(T) gravity for anisotropic sources
3 Model of anisotropic compact stars in f(T) gravity with Finch–Skea geometry
4 Conditions for physically realistic stellar model
 (i)At the boundary of a star (i.e. at \( r=b\)), the interior spacetime solution should be matched with the exterior Schwarzschild solution. For the continuity of the metric functions at the surface. One obtains,$$\begin{aligned}&D^2\left[ (B{}A\sqrt{(1{}\alpha )(1{+}Cb^2)})Cos\sqrt{(1{}\alpha )(1{+}Cb^2)}\right. \nonumber \\&\qquad \left. {+}(A{+}B\sqrt{(1{}\alpha )(1{+}Cb^2)})Sin\sqrt{(1{}\alpha )(1{+}Cb^2)}\right] ^2 \nonumber \\&\quad =\left( 1\frac{2M}{b}\right) \end{aligned}$$(26)$$\begin{aligned}&(1+Cb^2)=\left( 1\frac{2M}{b}\right) ^{1} \end{aligned}$$(27)
 (ii)The radial pressure drops from its maximum value ( at center ) to zero at the boundary, i.e.$$\begin{aligned} p_{r}_{ (r=b)} =0 \end{aligned}$$(28)
 (iii)
The density and pressure are positive inside a star, and at the center \(\rho (0)=\rho _0\), \(p_r(0)=p_{0}\), these are finite. For density this coincides with the null energy condition (NEC).
 (iv)
The model must satisfy the dominant energy condition (DEC) at the center, i.e. \(\rho _0> \mid p_0 \mid \).
 (v)
Inside the star the square of the sound velocity \(v^2 = \frac{dp}{d\rho } \leqslant 1\), to satisfy causality.
 (vi)
The gradient of the pressure and energydensity should be negative inside the stellar configuration, i.e., \(\frac{dp_r}{dr} < 0\), \(\frac{d\rho }{dr} < 0\) and \(\frac{dp_t}{dr} < 0\).
5 Physical analysis
5.1 Density and pressure of a compact object in f(T) gravity
5.2 Anisotropic behaviour
5.3 Stability
5.4 Mass–radius relation
Parameter values for different known compact objects with \(\alpha =0.4\) and \(\beta =1\)
Name of the star  Mass \((M_{\odot })\)  Radius (km)  A  B  C  EoS 

PSR J0348+0432  2.01  11  0.4729  0.3259  0.0096  \(p_r=\,44.876\rho ^{2} + 0.262 \rho 0.8\times 10^{4}\) 
HER X1  0.88  7.7  0.4696  0.5100  0.0085  \(p_r=\,25.128 \rho ^{2} + 0.157\rho 0.8 \times 10^{4}\) 
SAX J1808.4−3658 (SS1)  1.435  7.07  0.4667  0.2611  0.0238  \(p_r=\,23.989 \rho ^{2} + 0.328 \rho 2.2 \times 10^{4}\) 
SAX J1808.4−3658 (SS2)  1.323  6.55  0.4672  0.2664  0.0343  \(p_r=\,16.129 \rho ^{2} + 0.321 \rho 3.1\times 10^{4}\) 
Values of different physical parameters considering different values of the model parameter \(\beta \) for PSR J0348+0432
\(\beta \)  Observed mass \((M_{\odot })\)  Observed radius (km)  Radius from our model  \(\rho _0\;\;\;\;\;\;\)  \(\rho _b\;\;\;\;\;\;\;\;\;\;p_{r0}\;\;\;\;\;\;\;\;\;\;\frac{2M}{b}\le \frac{8}{9}\) 

0.5  \(2.01 \pm 0.04\)  =  16.32  0.00058  \(0.00008\;\;\; 0.00015\;\;\; 0.363\) 
1  =  11  11.02  0.00115  \(0.00034\;\;\; 0.00029\;\;\; 0.538\) 
1.5  =  =  9.00  0.00173  \(0.00067\;\;\; 0.00045\;\;\; 0.659\) 
2  =  =  7.875  0.0023  \(0.00108\;\;\; 0.00059\;\;\; 0.753\) 
2.5  =  =  7.1425  0.00288  \(0.0015\;\;\;\; 0.00075\;\;\; 0.830\) 
3  =  =  6.634  0.00346  \(0.00195\;\;\; 0.00089\;\;\; 0.893\) 
For different stellar objects it is evident that u will depends on the model parameters C, \(\beta \) and \(\beta _1\). In Table 2, we have obtained the values of radius for a fixed mass object considering different \(\beta \) values with \(\beta _1=0\). We have also determined the central density (\(\rho _0\)), radial pressure at the center (\(p_{r0}\)) and density at the boundary (\(\rho _b\)) of the star. From the table it is evident that the radius obtained from our model agrees well with the observational value when we consider \(\beta =1, \beta _1=0\) case. We note that for a given mass object the value of \(\beta \) lies between 0 and 3 (\(0<\beta <3\)), as \(\beta \) can not be negative for physically realizable models and also \(\beta \ge 3\) leads to \(\frac{2M}{b} >\frac{8}{9}\), which is not acceptable.
5.5 Equation of state
The variation of the energydensity and radial pressure are plotted in Figs. 1 and 2 from which one can predict possible EoS. Because of the complex form of the expressions of \(\rho \) and \(p_r\) it is not possible to obtain a known analytic relation between them. However a numerical analysis can be performed to predict the EoS. Here we obtain the best fitted relation between \(\rho \) and \(p_r\), the expressions so obtained for different compact objects have been listed in Table 1. In Fig. 13 both the linear and quadratic variation of \(p_r\) with \(\rho \) have been found to satisfy for PSR J0348+0432. It is predicted that the model permits both linear and quadratic fitting however the goodness of fit for the quadratic fit is better than the linear one as is evident from the figure also. Also it is noted that MIT bag model is not permitted here.
6 Comparative study of Finch–Skea star under f(T) and GR framework
Parameter values for different known compact objects for matter distribution following Chaplygin EoS for \( \alpha \ne 0\) (\(\beta =1\), \(\beta _{1}=0\) and \(\gamma = 1\))
Compact star  A  B  C  \(\xi \)  \(\zeta \) 

PSR J0348+0432  1.113  0.135  0.009  0.3  \(7.44\times 10^{8}\) 
SAX J1808.4−3658 (SS2)  0.883  0.119  0.034  0.3  \(5.5\times 10^{7}\) 
HER X1  2.661  0.086  0.008  0.3  \(3.24\times 10^{7}\) 
Parameter values for different known compact objects for matter distribution following Chaplygin EoS for \( \alpha = 0\) (\(\beta =1\), \(\beta _{1}=0\) and \(\gamma = 1\))
Compact star  A  B  C  \(\xi \)  \(\zeta \) 

PSR J0348+0432  0.670  0.072  0.010  0.3  \(7.44\times 10^{8}\) 
SAX J1808.4−3658 (SS2)  0.522  0.071  0.034  0.3  \(5.5\times 10^{7}\) 
HER X1  1.394  0.0511  0.008  0.3  \(3.24\times 10^{7}\) 
7 Anisotropic Finch–Skea stars with modified Chaplygin gas in f(T) gravity
7.1 Physical analysis of stellar model with Chaplygin gas
To study the physical behaviour of the energy density, pressure, anisotropy etc. inside the star the values of the model parameters A, B, C and \(\xi \) are obtained using the matching condition at the boundary together with the condition that the radial pressure vanishes as \(r=b\) for both \(\alpha \ne 0\) and \(\alpha = 0\) cases. The value of the parameter \(\zeta \) is then obtained for different stars of known mass–radius values for a physically viable stellar model which are tabulated in Tables 3 and Table 4 respectively.
7.2 Pressure anisotropy
8 Discussion
In this paper we consider a modified gravity described by \(f(T)=\beta T + \beta _1\), which is recently considered in describing issues in cosmology and astrophysics. Two different cases are considered, (1) considering only f(T) gravity with modified Finch–Skea geometry we predict the possible EoS of compact objects and (2) considering EoS described by the modified Chaplygin gas we determine the physical properties of compact objects. In the first case we found compact objects with quadratic EoS determined by fitting the model values of pressure and density numerically.
 (i)

In Fig. 1, we plot the variation of energydensity with different \(\beta \) values for a known compact object. It is found that for different \(\beta \), at the center the density of compact object in f(T) gravity is maximum and it monotonically decreases radially outward. In this case \(\beta \) plays the role of a scale and for a star of particular radius higher \(\beta \) values allows a star with more mass. It is also evident that \(\rho \) is independent of \(\alpha \), the modified Finch–Skea parameter.
 (ii)

In Fig. 2, we plot the variation of radial pressure for a known star. It is seen that the radial pressure is a monotonically decreasing function of r with its maximum value at the center of the stellar system. The value of the pressure increases with increasing \(\beta \) at a particular distance from the center of the star. The radial pressures for the isotropic (\(\alpha = 0\)) and anisotropic cases (\(\alpha \ne 0\) ) with a fixed value of the parameter \(\beta \) has been shown in Fig. 3 and it is seen that for the isotropic case the radial pressure for any value of r is greater than that of the anisotropic case except at the boundary, reaching a maximum value at the center of the star. In Fig. 4 we have plotted the radial variation of transverse pressure \(p_t\) and it is evident from the figure that the transverse pressure follows the same profile as the radial pressure. Figure 5 shows a variation of \(p_r\) and \(p_t\) for a particular value of \(\beta \). From the figure we see that both the pressures are equal at the center of the star and the difference between them increases with increasing r, reaching maximum at the surface of the star. This in turn implies a presence of anisotropy.
 (iii)

From Figs. 6 and 7, it is evident that energydensity and pressure gradient are negative for different \(\beta \) values.
 (iv)

From Eq. (33) we found that for nonzero values of \(\alpha \) the anisotropy depends on the parameter \(\beta \) which is a multiplicative constant in the torsional term of f(T) gravity. In Fig. 8 we have shown the radial variation of \(\varDelta \) for \(\beta >0\). It is evident that for \(\beta >0\), \(\varDelta \) is positive i.e, \(p_t>p_r\) which implies that the anisotropic stress is directed outwards, hence there exists a repulsive gravitational force that allows the formation of super massive stars. However for \(\beta <0\), the energydensity inside the star turns out to be negative also the causality of the speed of sound is violated which is not physically acceptable. If \(\alpha =0\), i.e for the Finch–Skea metric, the anisotropy vanishes in 4 dimensions, which corresponds to the isotropic case, although \(\alpha \ne 0\) leads to anisotropic compact star.
 (v)

We have shown the variation of the radial and transverse speed of sound in Figs. 9 and 10 respectively. It is also noted that the value of \(v_{st}^{2}v_{sr}^{2}\) is less than 1 complying the condition for a stable model of an anisotropic compact object, which is shown in Fig. 11. Thus the stability condition in the model described is satisfied.
 (vi)

From Fig. 12 it can be seen that the variation of mass function is regular inside the star and vanishes at the center. The compactification factor is also obtained in Eq. (35), which depends on C and f(T) model parameters for different stellar objects.
 (vii)

For a physically acceptable stellar model we have obtained the range of \(\beta \) value considering \(\beta _1=0\) for a given mass object. The range of allowed values is found to be \(0<\beta <3\) as can be seen from Table 2. If \(\beta \ge 3\) then the value of \(\frac{2M}{b}\) exceeds \(\frac{8}{9}\) which is not acceptable.
 (viii)

In Sect. 5.5, it is found that the EoS satisfies a nonlinear relation between \(p_r\) and \(\rho \). In Fig. 11 we plot the variation of \(p_r\) with \(\rho \) which is fitted with linear as well as quadratic equations. It is evident from the Fig. 13 that the quadratic fitting gives better acceptability for the model.
 (ix)

In Table 1, we tabulated the values of A, B, C and probable EoS for different known stellar objects considering their mass and radius from observations.
 (x)

In Sect. 7, we have considered the modified Finch–Skea metric and solved the field equations considering Chaplygin gas as matter distribution. The radial variations of \(\rho _{c}\), \(p_{rc}\), \(v_{tc}^{2}\) and \(v_{rc}^{2}\) are shown in Figs. 14, 15, 16, 17. It is evident from the curves that they are all well behaved inside the star.
 (xi)

In Sect. 5.2, it is seen that, stellar systems whose interior spacetime is described by the modified Finch–Skea geometry, the anisotropy depends on the modification parameter, \(\alpha \). If \(\alpha =0\) then anisotropy vanishes which is evident from Eq. (34). For Chaplygin gas model, the anisotropy is nonzero even in the case of generalized Finch–Skea metric (i.e. \(\alpha =0\)). The variation of anisotropic parameter, \(\varDelta \) inside the star is shown in Fig. 19 considering \(\beta >0\). It is evident that for positive \(\beta \) values anisotropy is positive and has a maximum value at the center of the star, which is opposite to that noted without Chaplygin gas which is an interesting result. It may be due to the introduction of modified Chaplygin gas in the stellar system under f(T) framework.
 (xii)

In Tables 3 and 4, the values of A, B, C are tabulated along with the parameters \(\xi \) and \(\zeta \) for different stellar objects considering modified Chaplygin Gas for both \(\alpha \ne 0\) and \(\alpha = 0\) respectively . It is noted that the modified Chaplygin gas plays an important role to understand the physical behaviour of a compact stellar object. We see that Finch Skea geometry with modified Chaplygin gas (MCG) permits an anisotropic star which is not possible in \(4\)dimensions without MCG.
Notes
Acknowledgements
AC would like to thank University of North Bengal for awarding Junior Research Fellowship. SD is thankful to UGC, New Delhi for financial support. The authors would like to thank IUCAA Resource Center, NBU for extending research facilities. The authors would like to thank anonymous referee for presenting the paper in its current form. BCP would like to thank DSTSERB Govt. of India (File No.: EMR/2016/005734) for a project.
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