Charged anisotropic strange stars in general relativity
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Abstract
The present paper provides a new exact and analytic solution of the Einstein–Maxwell field equations describing compact anisotropic charged stars satisfying the MIT bag model equation of state for quark matter. The model is obtained by assuming the Tolman–Kuchowicz spacetime geometry (Tolman, in Phys Rev 55:364, 1939; Kuchowicz, in Acta Phys Pol 33:541, 1968). Our stellar model is free from central singularity and obeys all the conditions for a realistic stellar object. The solution is smoothly matched with the exterior Reissner–Nordstrom spacetime in order to obtain the physical parameters of the system. An interesting phenomenon which arises in this model is the fact that the force due to the pressure anisotropy initially dominates the Coulomb repulsive force, nevertheless as the radius increases the electric force dominates the anisotropic one. This may be an additional mechanism required for stability and equilibrium against the gravitational collapse of the stellar object. Detailed analyses of the obtained model are also given with the help of graphical representations.
1 Introduction
At present, it remain a great challenge to obtain analytic solutions to Einstein field equations describing compact configurations e.g neutron stars, strange stars, white dwarfs and black holes, all of them representing the final stages of a star’s evolution. However, last decades the theoretical studies modeling these kind of objects have improved considerably. Specifically the development of fluid sphere models containing anisotropic matter distributions i.e. unequal radial and tangential pressure: \(p_{r}\ne p_{t}\). Of course, from the astrophysical point of view it represents a more realistic and intriguing scenario. The pioneering work by Bowers and Liang [3] about anisotropic spheres with uniform energy density suggested that anisotropy could also play an important role in describing the high redshift objects like quasars and could has a significantly affect on the physical parameters like maximum compactness, mass and radius of star. Later on it was shown that the inclusion of anisotropies within the stellar configuration play an important role in the stability and equilibrium. Heintzmann and Hillebrandt [4] 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 star. Moreover, Herrera and Santos [5] studied local anisotropy in selfgravitating systems supposing that an anisotropic model can be stable. On the other hand the studies by Gokhroo and Mehra [6] suggested that stability is improved for a positive anisotropy factor i.e. \({\varDelta }\equiv p_{t}p_{r}>0\), in distinction with the isotropic case, allowing it the construction of more massive and compact objects. Another way to enhance the stability of the system was analyzed by K. Dev and M. Gleiser submitting the object under small radial adiabatic oscillation when anisotropy is present [7], also they explain that for many compact stars with surface redshift \(Z_{s}>2\) can be described by assuming anisotropic matter distributions at the interior [8]. There is an abundant literature devoted to the study of the effect of local anisotropy on the global properties of relativistic compact objects [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72] (and references contained therein).
Bonnor’s [73] investigations on charged isotropic solutions and subsequent studies by Ivanov [74] showed that singularities can be avoided during the gravitational collapse, consequently the presence of a net electric charge improves the balance and stability of the system. It is worth mentioning that some solutions which do not meet the admissibility physical criteria become relevant after the inclusion of charge in them [75, 76]. Over the years several works available in the literature have been addressed the study of models including both anisotropy and electric charge [77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92]. In fact, as was pointed out early the presence of anisotropy and electric charge improve the stability and equilibrium of the configuration. The first one introduces a repulsive force (in the case \({\varDelta }> 0\)) that counteracts the gravitational gradient, while the second one does it due to the Coulomb force.
On the other hand, as mentioned earlier, it is not an easy task to solve Einstein’s equations analytically. In the case of isotropic uncharged fluid solutions, one has four unknown functions i.e. \(\{\rho ,p,e^{\nu },e^{\lambda }\}\) and in the case of anisotropic uncharged solutions one has five unknown functions that is \(\{\rho ,p_{r},p_{t}, e^{\nu },e^{\lambda }\}\) and three equations. However, in the case of Einstein–Maxwell equations one has six unknown functions i.e. \(\{\rho ,p_{r},p_{t},e^{\nu },e^{\lambda },E\}\) and four equations. Therefore, in order to solve this system it is necessary to give additional information. For example, assume a suitable form for the metric potentials and impose an adequate equation of state (EoS) e.g. \(p_{r}=f(\rho )\). Within this framework several authors have been considered the well known linear equation of state based on the MIT bag model [6, 14, 36, 44, 79, 80, 81, 82, 83, 85, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128]. From the theoretical point of view, the quark matter hypothesis put forward by Witten [129], has driven the study of an entirely new class of compact astrophysical objects composed of strange quark matter called strange stars. According to Alford [130], in the dense core of a neutron star there is sufficiently high density and corresponding low temperature to crush the hadrons into quark matter. In the MIT bag model [131] for strange stars, the quark confinement has been assumed to be caused by a universal pressure \(B_{g}\), called the bag constant. The studies by Farhi and Jafee [132] and Alcock et al [133] had shown that for a stable strange quark matter the value of the bag constant should be \(B_{g}\sim \)55–75 MeV/fm\(^{3}\). Besides the studies by Chamel et al. [134] on nucleonic EOSs BSk 19, BSk 20 and BSk 21, used the values of effective bag constant to be 78.6, 65.5 and 56.7 Mev/fm\(^{3}\), respectively. However, the datasets of CERNSPS and RHIC22 [135] show that a wide range of bag constant are permissible.
Following the above spirit the main motivation of this article is to develop some new analytical relativistic stellar models by obtaining closedform solutions of Einstein–Maxwell field equations. So, the outline of this work is: Sect. 2 presents the Einstein–Maxwell equations for anisotropic fluid spheres, regarding the MIT bag model EoS, Sect. 3 is devoted to the physical and mathematical analysis on the constant parameters in order to obtain a well behaved solution. In Sect. 4 we match the solution with the Reissner–Nordstrom spacetime, Sect. 5 we study the equilibrium, stability and energy conditions. Finally, Sect. 6 concludes the work.
2 The Einstein–Maxwell field equations
3 Bounds on the physical parameters
3.1 Regularity at centre (\(r=0\))
4 Junction conditions
The predicted radius for the strange star of Mass \(M=2.1\left( M_\odot \right) \) in Tolman–Kuchowicz space time via. MIT bag equation of state
Strange stars candidates  \(M\left( M_\odot \right) \)  R (km)  M / R  \(\frac{Q^2}{R^2}\)  a (\(\hbox {km}^{2}\))  b (\(\hbox {km}^{4}\))  h (\(\hbox {km}^{2}\))  \(B_g\) (\(\hbox {km}^{2}\))  C 

I  2.01  8.0  0.37059  0.0018796  0.02547  0.000295  0.011059  2.0002\(\times 10^{4}\)  0.063251 
II  2.01  7.1  0.41757  0.0391853  0.0409  0.000725  0.018409  2.43652\(\times 10^{4}\)  0.031871 
The central density (\(\rho _0\)), surface density (\(\rho _s\)), central pressure (\(p_0\)), surface redshift (\(z_s\)) and bag constant (\(B_g\)) for Strange star candidates
Strange stars candidates  \(\rho _0\,\)(g/cm\(^{3} \))  \(\rho _s\, \) (g/cm\(^{3}\))  \(p_0\,\) (dyne/cm\(^{2}\))  \(z_s\)  \(B_g \,\) (MeV fm\(^{3}\)) 

I  \( 4.103217\times {{10}^{15}}\)  \(1.079809\times {{10}^{15}}\)  \(9.072097\times {{10}^{35}}\)  0.95918  151.5457 
II  \(6.58899\times {{10}^{15}}\)  \(1.31536\times {{10}^{15}}\)  \( 1.582416\times {{10}^{36}}\)  1.21452  184.6062 
5 Some silent features of strange star models
5.1 Equilibrium under four different forces
It is observed from Fig. 4 that the system is in complete equilibrium under the foregoing mentioned forces. Inspection of the upper panel (corresponding to the star I) and the lower panel (corresponding to star II) shows that the force due to anisotropy initially dominates the electromagnetic force, nevertheless as the fractional radius r / R increases the electromagnetic force dominates the anisotropic force (it phenomenon occurs approximately at \(r/R=0.5\)). This change between these forces can be explained by the presence of a high electric field (approximately at \(r/R=0.5\)) of the compact object.
5.2 Energy condition
5.3 Stability via subliminal sound speeds
5.4 Effective mass–radius relation
The lower and upper bounds for the mass–radius ratio and the mass–radius relation for Strange star candidates
Strange star candidates  Lower bound \(\frac{Q^{2}\left( 18R^{2}+Q^{2}\right) }{2R^{2}\left( 12R^{2}+Q^{2}\right) }\)  Mass–radius ratio \(\frac{M}{R}\)  Upper bound \(\frac{4R^{2}+3Q^{2}+2R\sqrt{R^{2}+3Q^{2}}}{9R^{2}}\) 

I  0.00140965  0.370594  0.445697 
II  0.0293571  0.41757  0.470205 
5.5 Electric charge
6 Concluding remarks

It is observed from Figs. 1, 2 and 3 that the model is free from physical and geometrical singularities. Of course, both metric potentials are finite, positive and well behaved at the center of the star i.e. \(e^{\lambda (r)}_{r=0}=1\) and \(e^{\nu (r)}_{r=0}=C>0\) and are monotone increasing functions with increasing radial coordinate r towards the surface within the compact configuration. Furthermore, all the thermodynamic observables \(\rho \), \(p_{r}\) and \(p_{t}\) are monotonically decreasing functions with increasing radius, positive and well behaved everywhere inside the system.

Through out the stellar distribution the anisotropy factor is positive (i.e. \({\varDelta }>0 \Rightarrow p_{t} > p_{r}\)), it helps to construct a more massive and compact stellar structure.

The model satisfied simultaneously the standard point wise energy conditions that are required by normal matter, i.e. the NEC, WEC and SEC. Therefore, the energymomentum tensor is well defined everywhere inside the star, it can be corroborated in Fig. 5. The graphical representation of TOV equation (Fig. 4) shows that the stellar structure is in equilibrium under gravitational, ani sotropic, electric and hydrostatic forces. Where initially the aniso tropic force dominates the electric one, however as the radial coordinate grows the electric force finally dominates the anisotropic force. This implies that the surface layers are more stable (larger repulsive forces here) than the inner core layers. Therefore, the equilibrium of the compact object is enhanced.

For our model, causality condition is satisfied and stability through Abreu et al. criterion hold, representing a stable configuration.

The influence over the surface redshift \(Z_{s}\) and the mass–radius ratio u due to the presence of anisotropies and electric charge in the system are shown in Tables 2 and 3 respectively. The obtained values obtained here are in correspondence with the expected values for compact objects including charged anisotropic matter distribution. Additionally, Fig. 8 shows the trend of the surface redshift inside the star. As we can see it can not be arbitrary large due to the value of compactness (\(u=M/R\)) of the stars.

The values for the surface electric charge found in this model, are within the range of the values reported in previous studies [34, 146, 147, 148]. These values may be interpreted to represent the strangelet charge of strange stars made of color superconducting strange matter [145]. Furthermore, the electric field intensity \(E^{2}\) is well behaved in all points inside the configuration and it is vanishes at center as is expected. This can be seen in Fig. 9.
Finally, it can be concluded that an analytic solution to the Einstein–Maxwell field equations has been obtained, which meets all the requirements to be a physically and mathematically admissible solution representing a static, spherically symmetric spacetime described by a charged anisotropic en ergymomentum tensor.
Notes
Acknowledgements
S. K. Maurya acknowledge continuous support and encouragement from the administration of University of Nizwa. F. TelloOrtiz thanks the financial support of the project ANT1756 at the Universidad de Antofagasta, Chile.
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