Effect of electric charge on anisotropic compact stars in conformally symmetric spacetime

We obtain a new class of interior solutions for charged anisotropic stars which admits non-static conformal motion. The Einstein–Maxwell field equations are solved by taking a physically reasonable choice for the grr metric co-efficient and a suitable expression for charge density. From the analysis we have shown that there is a central singularity on space–time by calculating Kretschmann scalar though the central density and central pressure are finite. The emphasis are given on the causality condition and the stability analysis of the model. The masses and radii of the compact stars PSR J16142230, PSR J1903 + 327 and LMC X-4 obtained from the model are well match with the observed values provided by Gangopadhyay et al (2013 Mon. Not. R. Astron. Soc. 431 3216)


Introduction
To study the model of compact star is always an immense interest to the researchers since astronomers now believe that stars were the first large objects to form in the early universe. According to recent theoretical advances it is well known that at a very high density of the core 10 gm cm 15 3 -

(
) the pressure inside a fluid sphere shows anisotropic behavior [1], i.e., it has two components: radial pressure p r and tangential pressure p t . The existence of a solid core or by the presence of type 3A superfluid [2], different kinds of phase transitions [3], pion condensation [4] may occur anisotropy. A collective works devoted to the study of anisotropic spherically symmetric static general relativistic configurations are presented by Bowers and Liang [5]. Heintzmann and Hillebrandt [6] studied fully relativistic, anisotropic neutron star models at high densities and have shown that for arbitrary large anisotropy there is no limiting mass for neutron stars, but the maximum mass of a neutron star still lies beyond M 3 4  -. By assuming the condition of a vanishing Weyl tensor, Hernandez and Nunez [7] presented a general method for obtaining static anisotropic spherically symmetric solutions satisfying a nonlocal equation of state (EoS). Based on a particular form of the anisotropy factor, a class of exact solutions of Einsteins gravitational field equations describing spherically symmetric and static anisotropic stellar type configurations was obtained by Mak and Harko [8]. A variety of simple models for strange stars was constructed and compared by Avellar and Horvath [12]. Bini et al [12] studied the perturbation of white dwarf by using the 'effective geometry' formalism which was described as a self-gravitating fermion gas with completely degenerate relativistic EoS of barotropic type. Pagliara et al [12] studied the hydrodynamic simulation under the hypothesis that the conversion of a hadronic neutron star into a strange star is a combustion process and they also calculated the neutrino signal which is expected when a neutron star, decay into a strange quark star. Malaver [12] obtained a compact relativistic objects by considering Van der Waals modified EoS with polytropic exponent for anisotropic matter distribution. Petri [13] proposed a new class of solutions to the classical field equations of general relativity with zero cosmological constant and the proposed model has a sharp, non-continuous boundary of the matter-distribution, which is accompanied by a membrane consisting of pure tangential pressure. Numerical solutions of Einsteins field equation describing static spherically symmetric conglomerations of a photon star with an EoS p 3 r = was reported by Schmidt and Homann [14]. Chan et al [15] studied in detail the role of local pressure anisotropy in the onset of instabilities and they also showed that small anisotropy drastically change the stability of the system. Dev and Gleiser [16] proposed several exact solutions for anisotropic stars by taking constant density. We have extensively studied the charged and uncharged model of compact stars in some of our earlier works [17][18][19][20][21][22][23][24][25][26]. To find the exact solution, a more systematic approach is to assume conformal symmetry of the space-time. Conformal mapping is translated by the following relationship L g g y = x . Where L is the Lie derivative operator and ψ is the conformal factor. By assuming the supposition of spherical symmetry and the existence of a conformal Killing vector (CKV), Mak and Harko [27] found a relativistic model of strange quark star. Esculpi and Alomá [28] developed an anisotropic relativistic charged model by assuming the existence of a CKV in static spherically symmetric spacetime. Bhar et al [29] provide a new class of interior solutions for anisotropic stars admitting conformal motion in higher dimensional noncommutative spacetime by choosing a particular density distribution function of Lorentzian type as provided by Nazari and Mehdipour [30,31] under a noncommutative geometry. Anisotropic star model admitting conformal motion have been obtained by Rahaman et al [32,33]. A charged gravastar admitting conformal motion has been studied by Usmani et al [34] and Bhar [35] has generalized this result in higher dimension.
Inspired by these earlier work in the present paper we want to model an anisotropic charged compact star which admits a non-static conformal symmetry. Our paper is arranged as follows: in section 2, the field equations are presented; in section 3, exact general solutions are deduced using non-static conformal symmetries; in sections 4-8, the physical properties of the model are described. In the next section, we matched our interior solution to the exterior R-N line element and finally some conclusions are made in section 10.

Interior space-time
To describe the interior of a static and spherically symmetry object in the canonical coordinate x t r , , , where ν and λ being the functions of the radial coordinate 'r'. The Einstein field equations with cosmological constant can be written as We have assumed the matter distribution within the star is locally anisotropic and with some net electric charge. Therefore, the energy-momentum tensor is described by, where p , r r and p t represents the matter density, radial and transverse pressure of the fluid distribution. The quantities v m and c m are four-velocity and the unit spacelike vector in the radial direction that satisfies . Also  mn represents the electromagnetic stress-energy tensor defined as = mn m n n m represents the electromagnetic field tensor that satisfies the Maxwell's field equations given by where A μ is the electromagnetic four-potential, j m is the four-current density and 0 s is the proper charge density.
For a static fluid configuration, the non-zero components of the four-current density is j 0 and function of r only because of spherical symmetry. On using (6) we get where ¢ ( ) represents differentiation with respect to the radial coordinate r. From equations (10) and (11) the anisotropy factor Δ is obtained as,

Conformally symmetric charged anisotropic solution
To explore new solutions of Einstein's field equations we are searching for symmetry using CKV. Such method was adopted by Moopanar and Maharaj [36] to find conformal Killing symmetries in spherical space-times.
Recently, Radinschi et al [37] used non-static conformal symmetry for an anisotropic spherical configuration to study a classical model of electron. The non-static conformal Killing equation for (1) becomes where  x is the Lie derivative operator and ψ is the conformal factor. Also, the vector ξ generates the conformal symmetry and conformally mapped g mn onto itself along ξ. We follow Herrera et al [38,39] assumption that ξ is non-static for a static ψ and written as Using (1), (13) and (14) we get where A B C k , , , are all constants. To solve the above equations we have chosen A=0 and B=1 along with The behavior of e n and e l are shown in figure 1. Now using (18) and (17) are also plotted in figure 6 showing that they are 1  . The radial and transverse EoS are following quadratic behavior as shown in figure 7.
We can write the density and pressure gradients as Their behaviors with respect to radial coordinates is shown in figure 8.

Conditions for physical viability of the solutions
The following conditions are to be fulfilled by the solution in order to represent a physically viable configuration.     (v) For a stable anisotropic compact star, | must be satisfied [40].
(vi) Anisotropy must be zero at the center.

Properties of the solution
The central values of p p , r t , ρ and the Zeldovich's condition can be written as Here to get finite positive values of p rc and c r , we must have real and finite value of R.
Now the velocity of sound within the stellar object can be determine using (24) and (26) The solution satisfy the causality condition as the speed of sound is less than that of light, figure 11. The stability condition postulated by Herrera et al [40] i.e. v v t r 2 2 -| |should lies in between 0 and 1 which is again fulfilled, figure 12.
The relativistic adiabatic index is given by For a static configuration at equilibrium r G has to be more than 4/3 [15] which is also obeyed by our solution, figure 13.

Kretschmann scalar and singularity
Any singularity on space-time can be confirmed by using Kretschmann scalar whether it is a true singularity or coordinate singularity. It is defined as The Kretschmann scalar divergence at r=0, fugure 14 showing the space-time possesses a singularity at the origin. Other than the central singularity, our solution is regular everywhere. An interesting property of the solution can be seen from the values of central density and pressure that they possesses finite values even though its curvature diverges. This means that the solution has finite values of density and pressure at the singularity r=0.

Static stability criterion
In all modeling problems, it is required to check whether the solution (numerical or exact solution) is satisfying the necessary criterion i.e. static stability criterion [42,43]. In this criterion it is postulated that the total mass of the stellar structure must increase with respect to increase in central density   r <´belongs to stable region and vice versa. However, for increasing q, the stable range of c r increases, henceforth inclusion of some electric charge improve stability of stellar configurations.

Results and conclusions
In the article, we have explored a new charge anisotropic solution admitting non-static conformal symmetry.
We have assumed a time-dependent conformal symmetry to generate our solution, however, the obtained solution is a static or time-independent. All the physical parameters are well-behaved at the interior and possesses finite values at the center, however, at the center it contain a singularity with finite p c and c r as the Kretschmann curvature scalar diverges, figure 14. The resulting EoS and stability are strongly depends on the charge parameter q. For q=0, the EoS behaves almost linearly, however, when q increases the EoS start behaving in quadratic nature. Using the static stability criterion, the solution reveals that inclusion of more electric charge enhance the stability of the stellar configuration ( figure 7). Also inclusion of more electric charge supports more masses with a bigger radius. It is also seen that inclusion of more electric charge increase the compactness parameter M r 2 b within Buchdahl limit. In overall, the salient features of the solution can be summarized as: the solution holds all the energy conditions good, obeys causality condition, the relativistic adiabatic index is more than 4/3 (and therefore static), well-behaved except at r=0, stable with respect to Herrera et al [40] criterion and with respect to static stability criterion. Since the solution yields finite central density and pressure we used it to present some models of compact stars. Few models of compact stars are shown in table 1. The masses and radii of the modeled stars are well match with the observed values provided by Gangopadhyay et al [44].
The solution has distinct character as it contains a central singularity, however, possesses finite density and pressure at the center. The central singularity is in fact a geometrical or coordinate singularity. This can convince from the expression of Kretschmann scalar where one of the term contents e 2 n -. For this solution, the central value of e n is zero, hence K = ¥. Due to the solution possessing a central singularity, the high values of the surface red-shift may arise.