The General Theory of Relativity provides the mathematical platform for describing the physical world generated by highly compact astrophysical objects. According to Einstein’s relativistic theory, the existence of matter and its gravitational field causes to exist space-time gets curvature. When an incoherent matter like dust is contracted and condensed, a condition is reached where gas degeneracy pressure, thermal pressure (negligible with respect to gas degeneracy pressure) and gravitational pressure are in equilibrium. This equilibrium state forms a dense astrophysical object like neutron star, white dwarf, quark star, strange star etc. In order to understand the structural and physical properties of this dense astrophysical object, it is needed to find out the exact solution of Einstein’s relativistic field equation. Therefore exploration of anisotropic solution of Einstein’s relativistic field equation attracts the researchers to do work in this field. In 1916 Schwarzschild  solved Einstein’s field equations for describing the internal structure of a dense astrophysical object. He gave two solutions, one corresponding to external space time geometry and other corresponding to internal geometrical structure of fluid distribution by assuming constant matter energy density. But this solution has singularity at the center. Due to non-linearity and complexity of the differential equation, it is difficult to find the exact solution of these equations. Tolman  gave eight solutions for describing the internal structure and physical properties of super dense astrophysical object. The first solution corresponds to Einstein’s universe of static configuration having constant density and pressure. The second and third solutions correspond to Schwarzschild-de Sitter and Schwarzschild interior solutions respectively. The fifth and sixth solutions gave infinite constant density and pressure while the fourth and seventh solutions are physically acceptable. With passage of time, the number of non-singular solutions of Einstein’s field equation for theoretical description of super dense astrophysical objects in static case were made by various researchers.
Buchdahl  gave well known limit of the compactness for stable fluid distribution and limit of emission of radiation due to gravitational collapse of the distribution. This limit plays an important role for the stability of fluid distribution. Delgaty-Lake  examined 127 solutions and concluded that only 16 solutions were physically accepted against six physical properties namely non-singularity, isotropy in pressure, positive definiteness of central pressure and density, zero pressure at the surface, decreasing pressure and density with increasing radial distance. In addition to these properties, only nine solutions (Tolman, Patwardhan and Vaidya  , Mehra  , Kuchowicz  , Matese and Whitman  , Durgapals two solutions  , Nariai  , Goldman  ) gave monotonically decreasing speed of sound with increasing radial distance. Other solutions (Adler  , Heintzmann  , Finch and Skea  etc.) are not well behaved under these examining properties. Maurya and Gupta  , Pant  , Ivanov   , Pant et al.  gave physically accepted isotropic solutions of Einstein’s non-linear differential equations of relativistic field theory.
The radial pressure of fluid distribution may be different from the tangential pressure which may lead anisotropy in pressure. In the study of the properties of internal structure of a super dense astrophysical object, the local anisotropy is commonly used by the researchers in their solutions. The phase transition of fluid, existence of a solid core and other physical phenomena create local anisotropy in the pressure of the fluid. The surface red-shift of an astrophysical object, its stable mass and other physical properties may be affected by by anisotropy present in fluid distribution within that object. Combination of effect of anisotropy with equation of hydrostatic equilibrium investigates role of equations of state for fluid distribution.
Some researchers, Sah and Chandra  , Komathiraj and Maharaj  , Herrera et al.  , Thirukkanesh and Regel  , Chaisi and Maharaj  , Sunzu et al.  , Maurya and Gupta  and Bhar et al.  etc. gave physically accepted solutions of Einstein’s field equations for anisotropic fluid distribution. The researchers Sah and Chandra  , Whitman and Burch  , Tikekar  , Ivanov  , Gupta and Kumar  , Herrera et al.    , Tewari and Charan    , Tewari  , Ivanov     , Maurya and Gupta  etc gave remarkable work in relativistic astrophysics. The stellar model given in this paper is the contribution of the work related to charged fluid distribution, done by Sah and Chandra. In this work, we discuss physically accepted solutions for anisotropic fluid distribution in addition with energy conditions, variation of mass with central density, equation of state and mass-radius relationship in contest of some astrophysical objects, SAXJ1808.4-3658, HerX-1, 4U1538-52 and LMC X-4.
The whole work of this paper is divided into nine sections. The second section is comprised of Einstein’s non-linear differential equations of field theory. Besides this the physical parameters like matter-energy density, radial and tangential pressures, anisotropy, surface and central red shifts are expressed in this section. In third section the parameters of physically acceptable non-singular solutions are given. A non-singular physically acceptable solution of Einstein’s field equations given by Sah and Chandra  for anisotropic fluid distribution is incooperated in fourth section. The physical properties of the solution are given in fifth section. The matching of interior metric of the distribution with the exterior metric given by Schwarzschild is included in sixth section. In seventh section the tabular and graphical representation of physical parameters of our model of super dense star corresponding non-singular and physically accepted solutions are presented. The stability criteria are given in eighth section. In ninth section, some conclusions of whole work presented in this paper are given.
2. Einstein’s Field Equation of General Relativity
The relationship between gravity due to existing fluid material and geometry of space-time, contained in the Einstein’s non-linear differential equations for relativistic field theory is given by
where , the energy momentum tensor for spherically anisotropic fluid distribution is defined as
where is the proper density, and are radial and tangential pressures of an anisotropic fluid along and perpendicular to (time-like four-velocity vector) respectively, is the unit space like vector along radial vector and metric tensor such that
For spherically symmetric stable fluid distribution, the metric element can be expressed as
where and are metric coefficients depending on radial distance r only.
Incorporating Equation (2) and Equation (3) with Equation (1), we get the following equations
In the above equations, (’) denotes the derivative against radial distance r.
The redshift due to gravity of dense spherically symmetric fluid distribution is given by
which turns out the gravitational redshift ( ) for compact stellar object and it’s surface redshift ( ) as
where Schwarzchild parameter or compactness, , is the radius of
spherically symmetric stellar fluid configuration of mass .
3. Physical Acceptability Conditions for Stable Stellar Configuration
The non-singular solution of Einstein’s field equation must satisfy the following physically accepted conditions
1) There is no singularity in the solution interior and on the surface of the astrophysical object. For these metric coefficients and , central pressure and central density should be positive definite throughout the interior of the object i.e. , and .
2) The density, radial and tangential pressures should be maximum at the center and decreased monotonically on moving from center to the surface of the fluid object i.e.
a) at center and and for maximum value of , and . such that the radial pressure gradient, is negative for .
b) For monotonically decreasing density and pressures for .
3) At boundary radial pressure, shold be equal to zero while tangential pressure, may not be equal to zero i.e. pressure anisotropy is zero at the center i.e. . The pressure anisotropy should be increased with radial distance from the center of fluid distribution and at the surface it must be for zero surface radial pressure.
4) The null energy condition (NEC) , weak energy condition (WEC) , and strong energy condition (SEC) with and should be satisfied throughout within the fluid object.
5) The causality conditions and must be
satisfied throughout the stellar object. The speed of sound must be decreased monotonically on increasing the radial distance and increased with increasing
density i.e. or and or .
Thus the equation of state for highly dense astrophysical object indicates that the speed of sound should be decreased with increasing radial distance.
6) The realistic adiabatic index and should be positive and radial adiabatic index, should be greater than .
7) The gravitational red shift must have positive finite value and be decreased monotonically with increasing the radial distance.
8) The difference of the square of the radial speed of sound and tangential
speed of sound should lie between −1 and 0 for the matter within the object.
The method of solving the Einstein equations is to choose ansatze for the two metric functions. Then the three Einstein Equations (4)-(6) give expressions for the matter components , and . The two metric components are so related to allow simple forms for the physical variables. There is no integration in fact.
4. A Physically Acceptable Non-Singular Solution
The non-singular solution of Einstein’s field Equations (4) to (7) presented by Sah and Chandra  is given by
Here has dimension (length)−2 and , are dimensionless constants.
Using Equation (12) and Equation (13), Equation (4) gives radial pressure of the fluid inside the object as
Equation (12) and Equation (13) with Equation (5) give tangential pressure as
Equation (6) gives the matter energy density for our solution as
The red shift and anisotropy of astrophysical fluid distribution are given by
The radial derivatives of pressures and given by Equations (14) and (15) are
The radial derivative of matter energy density given by Equation (16) is
The radial derivatives of redshift and anisotropy given by Equation (17) and Equation (18) respectively are
5. Conditions for Model Parameters for Physical Acceptable Solution
For positive value of metric coefficient i.e. is possible if . The monotonically increasing nature of metric coefficients and with increasing radial distance r for suitable choice of constants , and are shown in Figure 1(a) and Figure 1(b). The values of pressure p, density and red shift at the center are given by
For positive values of pressure , density and redshift , , and .
From Equations (19) to (23), the radial derivatives of pressure, density, red shift and anisotropy at the center of fluid distribution are zero.
From Equation (19) and Equation (20), at the center of fluid distribution
Figure 1. (a): The variation of Metric Potential “ ”; (b): The variation of Metric Potential “ ” against .
For maximum values of radial and tangential pressures at the center, the conditions of model parameters are i.e. and i.e. .
From Equation (21)
which shows that for all values of parameters, the matter energy density is maximum at the center of the fluid distribution i.e. .
At the center, the equations of state for matter distribution are given by
The central equation of state should obey the conditions and which gives .
From Equation (19) and Equation (21)
and from Equation (20) and Equation (22)
The causality conditions and are satisfied at the center which gives the condition for model parameters .
From Equation (22) and Equation (23)
For maximum value of red shift at the center, i.e. . Since , the anisotropy is minimum at the center for all values of model parameters.
It is observed that for model parameters given by
the metric coefficients and increase monotonically with increasing radial distance r as illustrated by (Figure 1(a) and Figure 1(b)). The matter energy density reduces monotonically with increasing radial distance r as shown in the (Figure 2) while radial and pressures decreases monotonically with increasing radial distance r such that the radial pressure becomes zero at the surface of the fluid distribution (Figure 3(a) and Figure 3(b)). The pressure density ratios are always positive and less than one throughout the distribution (Figure 4(a) and Figure 4(b)). The red shift and anisotropy with zero central value increases with increasing radial distance r (Figure 5, Figure 6).
6. Model Parameters by Matching Conditions
For the stable astrophysical fluid distribution the interior metric must be matched with the Schwarzschild exterior metric given by
For this the metric coefficients must be continuous (First fundamental form) and differentiable (Second fundamental form) at the surface at and .
Figure 2. Variation of Density “ ” against .
Figure 3. (a): Variation of Radial Pressure against ; (b): Variation of tangential Pressure with .
Figure 4. (a):Variation of with ; (b): Variation of with .
where ; , Schwarzchild parameter. Equation (38) to
Equation (41) represent the model parameters obtaining from these boundary conditions.
7. Tabular and Graphical Representation of Physical Quantities
From Equation (38) to Equation (41) the model parameters , , and
with or for an astrophysical object
Figure 5. Variation of Red Shift with .
Figure 6. Variation of Anisotropy factor with .
SAXJ1808.4-3658 are 0.7734, −0.1979c, 0.9475c and −0.2089 respectively. The physical quantities estimated in our model for astrophysical objects SAXJ1808.4-3658, HerX-1, 4U1538-52 and LMC X-4 are illustrated by means of Table 1, Table 2 and Figures 1(a)-13(b) corresponding to respective model parameters given in Table 3.
With the help of model parameters for astrophysical objects SAXJ1808.4-3658, HerX-1, 4U1538-52, LMC-X-4, CenX-3, VelaX-1, PSRJ1614-2230 and PSRJ0348+0432 given in Table 3 with compactness , a super dense compact star model can be made which may be useful for further study of the different properties of various astrophysical objects.
8. Stability Criterion
In this section our aim is to determine the physical requirement of realistic solution featuring the stability of compact stellar or astrophysical objects.
Table 1. Variation of , , , , , and with for an astrophysical object SAXJ1808.4-3658 corresponding to parameters or .
Table 2. , , , , , and for an astrophysical object SAXJ1808.4-3658 corresponding to parameters or .
8.1. Energy Conditions
Our proposed model of anisotropic fluid satisfies the following energy conditions within the framework of general relativity.
Table 3. The model parameters for few well known compact stars SAXJ1808.4-3658, HerX-1, 4U1538-52, LMC-X-4, CenX-3, VelaX-1, PSRJ1614-2230 and PSRJ0348+0432.
Null Energy Condition (NEC)
Weak Energy Condition (WEC)
Strong Energy Condition (SEC)
8.2. Equilibrium of Different Forces
There are three kind of forces viz gravitational force , hydrostatic force and anisotropical force acting on a compact star which are in equilibrium.
The generalized Tolman-Oppenheimer-Volkoff equaion for equilibrium condition under these forces for anisotropic fluid distribution is given by
where is effective gravitational mass and is defined by Tolman-Whittaker as
Now TOV equation reduces to
Figure 7. (a): Variation of NEC with ; (b): Variation of WEC with ; (c): Variation of SEC with .
In view of Equation (45) and Equation (48) the three forces are given by
8.3. Causality Conditions
The causality condition states that the radial and tangential speeds of sound
should not be more than one i.e. and anywhere within
the stellar object. The graphical representations of radial and tangential speeds of sound with respect to in the Figure 9(a) and Figure 9(b) which shows that the causality conditions are obeyed throughout the stellar configuration. The condition for potentially stable anisotropic fluid distribution is
and is the condition for potentially unstable
anisotropic fluid distribution. In our relativistic stellar model the stability factor satisfies the condition for potentially stable anisotropic fluid distribution everywhere inside fluid sphere depicted by Figure 10.
8.4. Relativistic Adiabatic Index
The relativistic adiabatic index for anisotropic fluid distribution of an astrophysical object is given by
Figure 8. (a) Variation of three forces , and with ;(b): Variation of three forces , and with .
Figure 9. (a): Variation of with ; (b): Variation of with .
Figure 10. Variation of Stability factor with .
8.5. Harrison Zeldovich Novikov Static Stability Criterion
Harrison  Zeldovich Novikov  static stability criterion are given by
for stable fluid distribution,
for unstable fluid distribution,
In our model the mass of configuration in terms of central density is given by
Figure 11. (a): Variation of with ; (b): Variation of with .
The rate of change of mass of stellar configuration with respect to central density is always positive. Consequently our solution gives stable stellar configuration. The variation of masses of astrophysical objects SAXJ1808.4-3658, HerX-1, 4U1538-52 and LMC-X-4 with their central density is graphically represented in Figure 12.
We have given a well behaved analytic charge free solution for spherical and symmetric anisotropic fluid distribution given by Sah and Chandra  in addition with energy conditions, variation of mass with central density, equation of state and mass-radius relationship in contest of some astrophysical objects, SAXJ1808.4-3658, HerX-1, 4U1538-52, LMC X-4, CenX-3, VelaX-1, PSRJ1614-2230 and PSRJ0348+0432. Throughout the paper all the calculations, Table 1 and Table 2 are made for astrophysical object SAXJ1808.4-3658. The graphs are drawn for the compact star SAXJ1808.4-3658, HerX-1, 4U1538-52 and LMC X-4. It is concluded that the physical properties pressures, mass energy density and redshift with positive central value decrease monotonically with increasing radial distance and the causality condition is satisfied throughout within super dense fluid distribution.
By using suitable model parameters given in Table 3, we have calculated the mass and radius of above mentioned compact stars presented in Table 4 and it is clear that calculated values of mass and radius of these stars is well fitted with the observational data. We have also obtained the central density, surface density, central pressure, and surface redshift given in Table 5. The compactness factor is
Figure 12. Variation of mass with central density.
Table 4. The Observed and calculated values of mass and radius of few compact stars SAXJ1808.4-3658, HerX-1, 4U1538-52, LMC-X-4, CenX-3, VelaX-1, PSRJ1614-2230 and PSRJ0348+0432.
Table 5. The Central Density, Surface Density, Central Pressure and Surface Red Shift of compact stars SAXJ1808.4-3658, HerX-1, 4U1538-52, LMC-X-4, CenX-3, VelaX-1, PSRJ1614-2230 and PSRJ0348+0432.
Figure 13. (a): Variation of radial pressure with density against r; (b): Variation of tangential pressure with density against r.
always less than . The red shift is also satisfied the upper bound limit for the
It is found that the present model is very close to the observed data of a number of compact stars like SAXJ1808.4-3658, HerX-1, 4U1538-52, LMC X-4, CenX-3, VelaX-1, PSRJ1614-2230 and PSRJ0348+0432 and many more given by Elebert et al.  , Abubekerov et. al.  , Rawls et al.  , Demorest et al.  and Gangopadhyay et al.  . Thus the solution obtained by us is very helpful in constructing the models of super dense astrophysical object like Neutron stars, Strange star, quark stars and many more.