Possible Formation of Compact Stars in f(R,T) Gravity

Possible Formation of Compact Stars in Gravity

M. Zubair , G. Abbas Ifra Noureen ,
Department of Mathematics,
COMSATS Institute of Information Technology, Lahore, Pakistan.
Department of Mathematics,
COMSATS Institute of Information Technology, Sahiwal, Pakistan.
Department of Mathematics,
University of Management and Technology, Lahore, Pakistan.
mzubairkk@gmail.com; drmzubair@ciitlahore.edu.pkghulamabbas@ciitsahiwal.edu.pkifra.noureen@gmail.com

This paper constitutes the investigations regarding possible formation of compact stars in theory of gravity, where is Ricci scalar and is trace of energy momentum tensor. In this connection, we use analytic solution of Krori and Barua metric (Krori and Barua 1975) to spherically symmetric anisotropic star in context of gravity. The masses and radii of compact stars models namely model , model and model are employed to incorporate with unknown constants in Krori and Barua metric. The physical features such as regularity at center, anisotropy measure, casuality and well-behaved condition of above mentioned class of compact starts are analyzed. Moreover, we have also discussed energy conditions, stability and surface redshift in gravity.

Keywords: Compact Stars, Gravity.

1 Introduction

The late time evolution of stars influenced by strong gravitational pull has been a largely anticipated field in astrophysics and gravitational theories. It facilitates in investigations of diverse characteristics of gravitating sources via various physical phenomenons. Baade and Zwicky (1934) proposed the inception of massive compact stellar objects, establishing the argument that supernova may result in a small and super dense star. This eventually came in to reality in 1967, when Bell and Hewish (Longair 1994, Ghosh 2007) discovered pulsars that are highly magnetized and rotating neutron stars. So, in reality, we come across a fundamental revolutionary shift from normal stars to compact stars, with a wide range in the form of stars to neutron stars, quarks, dark stars, gravastars and finally black holes.

In our present work, we are more specifically interested in study of compact stars category. Generally the homogeneity of the spherically symmetric matter configuration is emphasized while theoretical modeling of a compact star, satisfying the Tolman Oppenheimer-Volkoff (TOV) equation. Ruderman (1972) was the first one to argue that the nuclear matter density becomes anisotropic at the core of compact object. Analytic solutions of the field equations for various static spherically symmetric configurations for anisotropic compatible to interior of compact stellar modeling have been obtained in numerous works (Maurya and Gupta 2012,2013 2014, Maharaj et al.2014, Pant et al. 2014a, 2014b). The pressure inside the fluid sphere disintegrates in two parts, namely radial and tangential pressure. It has been investigated that anisotropy gave rise to the repulsive force that assists to construct the compact objects. In this context, it is established in (Kamal et al.2012) that the Krori and Barua (henceforth KB)(1975) metric provides an effective and realistic approach in modeling of compact stars.

The numerical simulations can be taken into account to explore the characteristics of compact stars from integrated TOV equations, if equation of state (EOS) is known. Rahaman et al. (2012) extended the KB models by using the Chaplygin gas EoS and discussed their physical features. Mak and Harko (2004) established standard models of spherically symmetric compact stars via exact solution of the field equations. They determined the impressions of physical parameters such as energy density, tangential and radial pressure, concluding that these parameters remain finite and positive inside the stars. The anisotropic exact models for compact objects with a barotropic EOS are discussed in (2006). Hossein et al.(2012) studied the impact of cosmological constant on anisotropic compact stars.

General Relativity (GR) being fundamental theory of gravity is successful in weak field limit, but insufficient to explore the strong field. The expected description of GR in strong field regime can be done by its modifications. The modifications in Einstein-Hilbert (EH) action are induced to arrive at alternative theories of gravity. Among modified theories of gravity gravity being most elementary modification of GR is extensively studied in context of existence and stability of neutron stars and compact stars (Arapoglu et al.2012, Alavirad and Weller 2013, Astashenok et al. 2014, 2015, Yazadjiev et al. 2014, Kausar and Noureen 2014, Noureen et al.2015). Abbas and his collaborators (2014, 2015a, 2015b, 2015c) analyzed a class compact stars in GR, ,(where is torsion scalar) with different equation of state.

The issue of accelerated expansion of the universe can be explained by taking into account the modified theories of gravity such as gravity (Harko et al.2011). The gravity provides an alternative way to explain the current cosmic acceleration with no need of introducing either the existence of extra spatial dimension or an exotic component of dark energy. Harko et al.(2011) generalized gravity by introducing an arbitrary function of the Ricci scalar and the trace of the energy-momentum tensor . The dependence of may be introduced by exotic imperfect fluids or quantum effects (conformal anomaly). As a result of coupling between matter and geometry, motion of test particles is nongeodesic and an extra acceleration is always present. In gravity, cosmic acceleration may result not only due to geometrical contribution to the total cosmic energy density but it also depends on matter contents.

Soon after the origination of gravity, its cosmological and thermodynamic implications including the energy conditions and dynamical analysis were extensively discussed (Shabani and Farhoudi 2014, Harko and Lobo 2010, Harko 2010, Azizi 2013, Sharif and Zubair 2012a, 2012b, 2013, Jamil et al. 2012a, 2012b, Momeni et al. 2015a, 2015b, Momeni and Myrzakulov 2015, Barrientos and Rubilar 2014). However, the explorations regarding compact stars in gravity are yet to be done. Herein, we are interested to study the structure of a class of compact stars Model 1, Model 2 and Model 3 in gravity.

The modified action in is as follows(Harko et al. 2011)


where is matter Lagrangian and denote the metric tensor. Different choices of can be considered, each of which directs to a specific form of fluid. In our present work the viable model we have chosen is of following type


where is a function of Ricci scalar and is some positive constant value. We have analyzed above mentioned model with , being a positive scalar. In our discussion, we have considered the model which does not imply the direct non-minimal gravitational coupling between scalar curvature and trace of the energy-momentum tensor in the Lagrangian level. But there may be coupling between matter and geometry which becomes apparent in the study of thermodynamics (Sharif and Zubair 2012a). The can be considered as matter correction term to gravity and in this particular study we choose . The main reason behind the difference on cosmology in ordinary gravity and in the above model is the non-trivial coupling between matter and geometry. In our previous work (Sharif and Zubair 2012b), we have reconstructed some explicit models of this gravity for anisotropic universe and explored the phantom era of dark energy.

We have investigated that only for and , the energy density of all the considered models remains positive and energy conditions are only valid for these values of and . So, through out the analysis, we have used the above values of and , it will not be mentioned again explicitly.

This paper is arranged as: In the following section anisotropic matter distribution and expressions physical parameters for energy density , radial and tangential pressure are established. Section 3 constitutes the analysis of physical features of compact stars and their stability analysis. In the last section, we have summarized our results.

2 Anisotropic Matter Distribution in Gravity

The line element for particular spherically symmetric metric describing the compact star stellar configuration is


where , (Krori and Barua 1975), , and are arbitrary constant that will be calculated by using some physical assumptions. The above set of functions are introduced to arrive at singularity free structure for compact star. Clearly, this set of functions leads to non singular density and curvature setting.

Taking and upon variation of modified EH action in (1.1) with respect to metric tensor , the following modified field equations are formed as


where , and denotes the usual matter energy momentum tensor that is considered to be anisotropic, is given by


where , and denote energy density, radial and transverse stresses respectively. The four velocity is denoted by and to be the radial four vector satisfying


When , the expression for , and can be extracted from modified field equations as follows


Here and prime denotes the derivatives with respect to radial coordinate. Substituting Eq.(1.2) in Eqs.(2.7)-(2.9) and inserting value of Ricci scalar in the form of metric functions together with the KB metric coefficients, we arrive at


The Schwarzschild solution is considered to be most suitable choice for matching conditions in exterior regime (Noureen and Zubair 2015, 2014, Goswami et al.2014 Cooney et al.2010, Ganguly et al. 2014). The interior metric of the boundary surface will be the same for the internal or external geometry of the star. Herein, the exterior metric of the star is described by the Schwarzschild solution, given by


Smooth matching of the interior metric (2.3) to the vacuum exterior solution at the boundary surface , yield


where superscript and stands for interior and exterior solutions. Matching of the interior and exterior spacetime leads to following


The values of the constants and are evaluated by using approximate values of and (Lattimer and Steiner 2014, Li et al.1999) of the considered compact stars, provided in the table 1 . The compactness of a star can be defined by and surface redshift can be determined by using the result . The values of for the considered models have been given in table 1.

Model 1 0.88 7.7 0.168 0.006906276428 0.004267364618
Model 2 1.435 7.07 0.299 0.01823156974 0.01488011569
Model 3 2.25 10.0 0.332 0.01090644119 0.009880952381
Table 1: Approximate Values of the model parameters for considered compact stars

3 Physical Analysis

This section covers the physical constraints required for interior solution, incorporating anisotropic behavior, matching and energy conditions together with the stability analysis of considered compact stars.

3.1 Anisotropic Behavior

Prior to the discussion of anisotropy measure, we discuss the evolution of energy density and anisotropic stresses and respectively, shown in Figures 1-3.

Figure 1: Evolution of energy density with radius
Figure 2: Variation of radial pressure with radius
Figure 3: Evolution of tangential pressure with radius

Radial derivative of equation (2.7) leads to the following expression

and also Eq.(2.8) reveals that , depicting decrease in energy density and radial pressure with increasing radius of the compact object, well supported by the results shown in Figures 4 and 5. Maximality of central density and pressure is achievable at , indicating

Figure 4: Behavior of with increasing radius .
Figure 5: Evolution of with increasing radius .

Using the EOS and , we obtain the following form of EOS parameters


It is interesting to mention here that the EoS parameters depend on radius rather than a constant quantity as in ordinary matter distribution. This non-constant behavior of EoS parameters is constituted by usual matter and exotic matter contributions (see Fig. 6 and 7).

Figure 6: The evolution of radial EoS parameter across stars.
Figure 7: The evolution of tangential EoS parameter across compact stars.

The measure of anisotropy , for considered model takes following form


Figure 8 describes the evolution of anisotropy measure, i.e., corresponds to the outward drawn anisotropy and its directed inward when . In our model, we find that for the different compact stars as shown in Figure 8. It can be seen from Fig. 8 that at most of the points for compact stars Model 1 and Model 3, indicating that that a repulsive anisotropic force occurs, allowing the construction of more massive distributions. In our model anisotropy measure for Model 2 decreases with the increase in radius and becomes negative beyond .

Figure 8: Anisotropy measure for compact stars

3.2 Energy Conditions

Energy bounds are of significant importance including null energy condition (NEC), weak energy condition (WEC), strong energy condition (SEC) and dominant energy condition (DEC), defined as

The considered anisotropic sphere satisfy the energy conditions, exhibited graphically in Figure 9 for compact star Model 1.

Figure 9: Energy conditions for Model 1

3.3 Causality Conditions and Stability Analysis

The radial and transverse sound speeds denoted by and should be less than speed of light i.e., , , where and . We plot the evolution of radial and transverse sound speeds for compact stars and found that above mentioned conditions hold, as shown in Fig. 10 and 11.

Figure 10: Plot of varying with radius .
Figure 11: Plot of varying with radius .

One can undergo with the stability analysis of compact objects considering sound speeds (Herrera 1992, Herrera and Barreto 2013, Herrera and Santos 1997, Herrera et al.2008). The the potentially stable and unstable regimes can be estimated by considering the difference of the sound propagation within the matter distribution. The variation of of different strange stars is shown in Figure 12. Figure 12 shows that difference of the two sound speeds, i.e., retain similar sign within the specific configuration and it satisfies the inequality . Thus, our proposed compact stars models are stable.

Figure 12: Plot of for anisotropic compact stars.

4 Conclusion

The modified theory of gravity can provide the explanation of accelerated expansion of universe. The modified gravity theories (such as the f(R,T) gravity) have been the center of attention for many researchers in the recent past, because this type of theories seems to provide a viable explanation for dark energy. The conformal relation of to general theory of relativity with a self-interacting scalar field has been examined (Astashenok et al. 2014).

This paper deals with the interior solutions for anisotropic fluids, which have been used to model compact stars in the context of modified gravity theory . The modeling has been completed by taking compact stars as anisotropic in gravity. For the exact solution of the governing differential equations, we have used the Krori and Barua (1975) form of the metric function, i.e., , , , and are arbitrary constant that have been calculated by using some matching conditions. The smooth matching of interior and exterior regions of a star lead us to express the unknown constants in terms of masses and radii of the compact stars. Using the observed values of the masses and radii of the compact stars, we get the values of the model constants that are used to discuss the nature of the stars. For the calculated values of the constant, we found that the energy density, radial and transverse pressure decreases for given class of compact strange stars. For the particular choice , we have found that EOS parameters behave like normal matter distribution. On the basis of this fact, we may conclude that compact stars are composed of ordinary matter and effect of term. The regularity analysis of the proposed model implies that density and pressure are regular every where and attain the maximum value at the center. Thus radial pressure and matter density have maximum values at the center and it decreases from the center to the surface of the star, where density becomes constant and pressure reduces to zero.

It has been found that the anisotropy will be directed outward when this implies that . We have found that for compact stars Model 1 and Model 3, indicating that that a repulsive anisotropic force occurs, allowing the construction of more massive distributions. While the anisotropy measure for Model 2 decreases with the increase in radius and becomes negative beyond . The the potentially stable and unstable regimes have been estimated by considering the difference of the sound propagation within the matter distribution. The compact stars remain potentially stable in the regions where difference of radial and sound speeds remain positive. It is found that for all three considered compact stars. So, our considered model is potentially stable in gravity, as shown in Fig. 12.

We would like to mention that TOV equations for modified theories of gravity like (Momeni and Myrzakulov 2015), non-local gravity (Momeni et al. 2015) and (Moraes et al.2015) have been studied analytically and numerically.

5 Conflict of Interest

Authors declare that there is no conflict of interest regarding the publication of this paper.


We are thankful to the unknown referee for his/her deep study of the paper and fruitful comments, that have improved the paper a lot.


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