Modeling Anisotropic Stars Obeying Chaplygin Equation of State
Abstract
In this work we provide a framework for modeling compact stars in which the interior matter distribution obeys a generalised Chaplygin equation of state. The interior geometry of the stellar object is described by a spherically symmetric line element which is simultaneously comoving and isotropic with the exterior spacetime being vacuum. We are able to integrate the Einstein field equations and present closed form solutions which adequately describe compact strange star candidates like Her X1, RX J 185637, PSRJ 16142230 and SAX J1808.43658.
Keywords:
General Relativity, Anisotropy, Compact star, Chaplygin equation of state∎
e1email:piyalibhar90@gmail.com \thankstexte2email:megandhreng@dut.ac.za \thankstexte3email:rsharma@associates.iucaa.in
1 Introduction
The search for exact solutions of the Einstein field equations has generated a rich field of models describing relativistic compact objects. Since the pioneering work of Schwarzschild who obtained the first interior solution describing a uniform density sphere, the modeling of relativistic stars has moved from the regime of toy models to sophisticated, realistic stellar structures. With the discovery of pulsars, neutron stars and strange stars there was a need to obtain relativistic analogues of Newtonian stars, particularly when the densities of the stellar material was of the order of gm cm. The simplistic model of a static uniform density star has been generalised to include the effects of pressure anisotropy, electric charge, scalar field, dark energy and the cosmological constant, on the gross physical properties of compact objectsrayso (); ma1 (); ma2 (). Models of relativistic fluid spheres have also been obtained within the framework of higher order theories of gravity including the RandallSundrum brane scenario, EinsteinGaussBonnet gravity and Lovelock formalismssud1 (); sud2 (); abbas (); nare (). In order to close the system of equations governing the gravitational and thermodynamical behaviour of bounded objects, various techniques were employed by researchers working in this field of study: (i) imposition of symmetry, (ii) adhoc assumptions of the gravitational potentials, (iii) specific choices of the falloff behaviour of the pressure, density or the anisotropy, to name a fewdev1 (); dev2 (); iv1 (). To construct a stellar model, a physically motivated route, in general, is to impose an equation of state which relates the pressure as a thermodynamical function of the density, ie., . Most of the earlier works were centered on imposing a linear equation of state of the form where is a constant. This was later generalised to , where and is the surface densitysm1 (). The conditions were relaxed by allowing for anisotropic pressure. Note that works in fundamental particle physics led to the MITbag model which hinged on an equation of state of the form where is the Bag constant. The linear equation of state was further generalised to the quadratic equation of state of the form sifiso (). One of the first successful attempts to obtain a generalisation of the Newtonian polytrope was achieved by Buchdahlbuch () in which he obtained a pseudorelativistic version of the LaneEmden polytrope of index . Herrera and Barreto presented a general formalism to generate relativistic polytropes with anisotropic pressure in Schwarzschild coordinatespoly (). Their findings also prompted further investigations into the origins of anisotropy, cracking in relativistic stellar models and stabilityazam (). In order to finetune these models with observations, some researchers employed a mixed polytrope equation of state in which two or more species of particles made up the stellar fluid. The inclusion of charge within the stellar core led to a plethora of static stellar models in which the role of the electromagnetic field on the stability, massradius ratio and redshift was demonstratedr1 (); ra1 (); ra2 ().
Issues surrounding the black hole horizon paradox necessitated the search for alternative models of black holes free of horizons. The gravastar model was first proposed by Mazur and Mottola mazur () which sought to address many of the problems encountered during the final stages of gravitational collapse. Dynamically the model hinged on the phenomenon that during the latter epoch of gravitational collapse spacetime itself would undergo phase transitions which would halt collapse. The emerging picture of a gravastar was that of a layered composite: a de Sitter interior filled with constant positive (dark) energy density featuring an isotropic negative pressure . This layer is then connected via three intermediate layers to an exterior vacuum Schwarzschild solution. The intermediate relatively thin shell is composed of stiff matter (). Stability of the composite profile is achieved by utilising two infinitesimallythin shells endowed with surface densities and surface tensions . An interesting model was proposed by Usmani et alra2 () in which they generalised the MazurMottola gravastar picture to include charge. In addition, the interior of the gravastar admitted conformal motion. They were able to show that charged interior de Sitter void must generate the gravitational mass. This mass is accountable for the attractive force that counterbalances the electromagnetic repulsion due to the presence of charge during the collapse processrahaman1 (). In a more recent model, Banerjee et albrig () presented a Braneworld generalisation of a gravastar admitting conformal motion. Motivated by the existence of dark energy, Lobo and coworkerslobo1 () have proposed stellar models, the socalled ‘dark stars’ in which the equation of state of is of the form in which . It has been proposed that in the phantom regime (), the extremely high pressures may invoke a topological change rendering the dark energy star to a wormhole. An interesting proposal regarding dark energy and dark matter is treating them as different manifestations of a single entity. This proposal leads to the Chaplygin gas model in which the equation of state derives from string theory. Various applications of the Chaplygin gas model have been pursued in order to account for cosmological observations such as acceleration of the cosmic fluid and structure formation. The Chaplygin equation of state has been subsequently modified to a more generalized Chaplygin gas equation of state. The generalised Chaplygin equation of state has been employed to model dark stars which are remnants of continued gravitational collapse. The idea here is that the dark energy provides sufficient repulsion to halt collapse leading to stable bounded configurations free of horizons and singularitiesp1 (); p2 ().
This paper is structured as follows: In section we introduce the field equations necessary for the modeling of a spherically symmetric star within the framework of general relativity. In section . we present a particular solution describing the interior of the star in which the matter content obeys a generalised Chaplygin equation of state. The junction conditions required for the smooth matching of the interior spacetime to the exterior Schwarzschild solution are worked out in section . A detailed physical analysis of the geometrical and thermodynamical behaviour of our model is presented in section . We discuss the stability, energy conditions and massradius relation in sections , and , respectively. We conclude with a discussion of our results in section .
2 Spherically symmetric spacetime
We consider a model which represents a static spherically symmetric anisotropic fluid configuration obeying a generalised Chaplygin equation of state. The interior spacetime is described by a spherically symmetric line element which is simultaneously comoving and isotropic
(1) 
where and the metric functions, and are yet to be determined. For our model the energymomentum tensor for the stellar fluid is
(2) 
where , and are the proper energy density, radial pressure and tangential pressure, respectively. The fluid four–velocity is comoving and is given by
(3) 
The Einstein field equations for the line element (1) are
(4) 
(5) 
(6) 
where primes denote differentiation with respect to the radial coordinate . We have utilized geometrized units in deriving the above system of equations in which the coupling constant and the speed of light are taken to be unity. The mass of the spherical object is given by
(7) 
where is an integration variable. In order to close the system of equations, we assume that the interior matter distribution obeys a generalised Chaplygin equation of state of the form
(8) 
where and are positive constants. Substituting (4) and (5) in (8) we obtain
(9) 
where
(10) 
(11) 
On integrating (9) we obtain
(12) 
where
(13) 
and is a constant of integration. Therefore, the line element (1) can now be written as
(14) 
where is given in (13). Hence, any solution describing a static spherically symmetric anisotropic matter distribution obeying a generalised Chaplygin equation of state in isotropic coordinates can be easily determined by a single generating function .
3 Generating solutions
In order to close the system of equations several choices for can be made. It is interesting to note that the choice of the metric potential determines the gravitational and thermodynamical behaviour of the model. Hence the choice of must satisfy all the requirements for a realistic stellar model. Recent work by Naidu and Govenderng2016 () have shown that the endstate of gravitational collapse resulting from a dynamically unstable static core is ‘sensitive’ to the choice of the initial metric functions. They show that for the same but with two distinct initially static cores; (i) vanishing radial pressure within the static configuration and (ii) uniform density interior, the final outcome of dissipative collapse leads to very different temperature profiles. Following Govender and Thirukkaneshgov1 () we utilise the physically motivated choice for as
(15) 
where and are constants. One can easily verify that the gravitational potential
in (15) satisfies the regularity conditions, = constant and at the
origin. The same expression of was previously utilized to model compact objects in curvature coordinates by Schwarzschild sch (), Einstein ein () and de
Sitter des () and more recently in comoving coordinates by Govender and Thirukkanesh gov1 () and Thirukkanesh et al. thir ().
With this choice of , we obtain from (12)
where is a constant of integration.
Subsequently, the field equations yield
(16)  
(17)  
(18) 
where, (i=1,2,…6) are given by
We define the anisotropic factor as
(19) 
which is repulsive in nature if and attractive if .
4 Matching Conditions
In this section we match the interior spacetime to the exterior spacetime described by the exterior Schwarzschild solution in comoving isotropic coordinatesbonnor1 ()
(20)  
where is the mass within a sphere of radius . Matching of interior metric (1) and exterior metric (20) at the boundary leads to the constraints
(21)  
(22) 
where
(23) 
The condition (21) imposes the following restriction on the constant of integration
The condition (22) implies
(25) 
which imposes a restriction on the parameters and which can be determined if we specify the radius of the sphere.
To examine the behabiour of the model parameters like matter density, radial and transverse pressure etc. we assume , , and . By using the matching conditions together with , we obtain the constant for a star of radius km. The mass of the stellar configuration turns out to be which is very close to the observed mass of the strange star candidate Her X1rawls ().
5 Physical Analysis
We are now in a position to discuss the the physical features of the model generated in the preceding section. In order to describe a realistic stellar structure our model must satisfy the following physical requirements :

Regularity of the gravitational potentials at the origin:
In our model, which are constants and at the origin , which indicates that the gravitational potentials are regular at the origin.

Positive definiteness of the energy density and pressure at the centre:
Since , the energy density is positive and regular at the origin. We also have . To ensure that the radial pressure is positive at the center we must have .
Moreover, i.e., the energy density is a decreasing function of .
We also note that , which implies that is a decreasing function of .

Continuity of the extrinsic curvature across the matching hypersurface, :
Continuity of the extrinsic curvature across the matching hypersurface, yields
(26) which gives
which is finite for appropriate choice of parameters and .

Ratio of trace of stress tensor to energy density :
Fulfillment of the requirement that the ratio of trace of stress tensor to energy density should decrease radially outward is shown graphically in Fig. (4).

Velocity of sound:
For causality to be obeyed the radial and transverse velocities of sound should be in between []. The radial velocity and transverse velocity of sound can be obtained as
(27) (28) Due to the complexity, we illustrate the causality conditions with the help of graphical representations. Fig. (6) and Fig. (7)clearly show that and everywhere within the stellar configuration.

Stability:
Following Heintzmann and Hillebrandthein (), a model of anisotropic compact star will be stable if everywhere within the stellar interior where the adiabatic index is defined as
(29) Fig. (8) shows that values of the adiabatic index which clearly indicates that the particular configuration developed in this paper is stable.

Energy conditions:
A realistic star should satisfy the energy conditions namely, the Weak Energy Condition (WEC), Null Energy Condition (NEC) and Strong Energy Condition (SEC) as given below:
(30) (31) (32) For the specific stellar configuration developed here, validity of the inequalities (30)(32) have been shown with the help of graphical representations in Fig. (10).
6 Massradius relation
The mass function in our model is obtained from (9) as
(33) 
The profile of the mass function is shown in Fig. (9). Since at we have , it implies that the mass function is free from any central singularity.
Buchdahlbuchdahl59 () obtained an upper bound on the mass to radius ratio i.e., compactness of a relativistic star of a compact star such that . In our model, we have
(34) 
The compactness for different compact star models are given in Table 2. The table shows that compactness of configurations are wihin the Buchdahl limitbuchdahl59 ().
We have also determined the surface redshift using the formula
which for our model turns out to be
(35) 
The values of the surface redshift parameter for different stellar configurations are given in Table 1. For an isotropic star, in the absence of a cosmological constant, Buchdahlbuchdahl59 () and Straumann stra () have shown that . Böhmer and Harkobh () showed that for an anisotropic star, in the presence of a cosmological constant, the surface redshift can take a much higher value . The restriction was subsequently modified by Ivanov ivanov () who showed that the maximum permissible value could be as high as . In our case, we have for different compact star models developed in this paper.
Compact Star  (km)  a  b  H  K  () 

Her X  1  6.7  1.3997  0.009  0.294  0.00012910  0.789 
RX J 185637  6.006  1.6859  0.024  0.21  0.00020280  0.904 
PSRJ 16142230  11.3664  2.0952  0.0138  0.29  0.000022416  1.86 
SAX J1808.43658  7.07  2.1015  0.036  0.29  0.000149572  1.157 
Compact Star  central density  surface density  central pressure  

gm cm  gm cm  dyne cm  
Her X  1  0.1737  0.2379  
RX J 185637  0.2219  0.3409  
PSRJ 16142230  0.2413  0.3905  
SAX J1808.43658  0.2414  0.3906 
7 Discussions
In this paper we have presented a new model of a compact star in isotropic coordinates which is free from central singularity with the exterior being the vacuum Schwarzschild spacetime. To solve the Einstein field equations we have employed the modified Chaplygin equation of state which is inspired by the current observation of the expanding universe and its connection to the existence of dark energy. By considering the observed radius of the compact star Her X1 as an input parameter (we have assumed km), we have analyzed physical viability of our model. We note that the metric coefficients are free from any singularity. The variations of , , are plotted in Fig. (1), (4), (2) and (3), respectively which clearly show that matter density, radial and transverse pressure are positive inside the stellar interior and they are monotonically decreasing functions of radial coordinate. At the boundary of the star the matter density and transverse pressure are nonnegative and the radial pressure vanishes as expected. From the plot of (Fig. (4)) we note that it is nonzero and monotonically decreasing from the center towards the stellar surface. The anisotropic factor is plotted against in Fig. (5). Since , the anisotropic factor is repulsive in nature in our model which is a desirable feature of a compact stargm (). Moreover, at the center of the star vanishes which is also an essential feature of a realistic star. In order to investigate the relevance of our model in the study of compact stars, we have considered a number of compact stars, namely Her X1, RX J 185637, PSRJ 16142230 and SAX J1808.43658 and showed that for the estimated radii the star’s masses determined from our model are very close to the observed masses (see Ref. maurya (); tj ()). This leads naturally to the proposition that the solution obtained in this paper can be used as a viable model for describing ultracompact stars.
Footnotes
 journal: Eur. Phys. J. C
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