A completely deformed anisotropic class one solution for charged compact star: a gravitational decoupling approach
Abstract
In this article, we have investigated a new completely deformed embedding class one solution for the compact star in the framework of charged anisotropic matter distribution. For determining of this new solution, we deformed both gravitational potentials as \(\nu ~\mapsto ~\xi +\alpha \, h(r)\) and \(e^{-\lambda } \mapsto ~e^{-{\mu }} + \alpha \,f(r)\) by using Ovalle (Phys Lett B 788:213, 2019) approach. The gravitational deformation divides the original coupled system into two individual systems which are called the Einstein’s system and Maxwell-system (known as quasi-Einstein system), respectively. The Einstein’s system is solved by using embedding class one condition in the context of anisotropic matter distribution while the solution of Maxwell-system is determined by solving of corresponding conservation equation via assuming a well-defined ansatz for deformation function h(r). In this way, we obtain the expression for the electric field and another deformation function f(r). Moreover, we also discussed the physical validity of the solution for the coupled system by performing several physical tests. This investigation shows that the gravitational decoupling approach is a powerful methodology to generate a well-behaved solution for the compact object.
1 Introduction
It is well known that the astrophysical compact objects are formed due to the gravitational collapse. These compact stars belong to three different groups like neutron stars, white dwarfs, and black holes. The arrangement of these stars is constructed according to their internal structure. The black hole is the densest stars in the universe. Till now, it is an open problem in astrophysics to know about the behavior, nature and exact configuration of these astrophysical compact objects which are more compact than the normal compact objects, therefore in recent days, it is still an active field of research. Based on the hypothesis of strange matter, the strange quark matter could be more stable than nuclear matter. Therefore, the neutron stars must essentially be composed of pure quark matter. The current expansion in a cosmological investigation has explained gradually the origin and distribution of substance and growth of compact objects in the Universe. Few of their assets, such as rotation frequencies, masses, and emission of radiation are computable. The measurements of the essential parameters decide the nature of compact objects which is still an observational challenge because these properties are not directly associated with the observations, such as the internal configuration of masses and radii that need the growth of hypothetical models. In view of theoretical computation, the mass and radii are calculated by solving the hydrostatic equilibrium equation that gives an equilibrium condition between the hydrostatic and gravitational force. In the case of Einstein general relativity, the Tolman–Oppenheimer–Volkoff (TOV) equations can represent the equilibrium condition for a spherical compact object. However, equation of state is necessary for describing the complete structure of these spherical compact objects. In the modern days, the experimental data shows the existence of such astrophysical objects which are observed at very high densities [1], and few of the astrophysical objects like Her X-1 (X-ray pulsar), PSR 0943+10, 4U 1820-30 (X-ray burster), 4U 1728-34, RX J185635-3754 (X-ray sources), and SAX J 1808.4-3658 (millisecond pulsar) extremely favor for the probability that these compact objects could essentially be strange stars. In further arguments, there is no any resilient confirmation to establish the mechanism for the understanding of astrophysical compact objects. According to the theoretical investigations, such compact objects are composed of a perfect fluid [2]. Mostly, the bag equation of state (EOS) and polytropic EOS (\(p=k\,\rho ^{\gamma }\)) have been extensively applied to describe less compact star and a white dwarf [3, 4, 5]. In this connection Herrera and Barreto [6] conveyed a general study on polytropic relativistic stars with anisotropic pressure while polytropic quark star models have been studied by Lai and Xu [7]. From a theoretical point of view, the impact of anisotropic compact objects was first introduced by Bowers and Liang [8]) and another study headed by Ruderman [9] revealed that nuclear matter may be composed of anisotropic pressure at a very high density which is the order of \(10^{15} \,\mathrm{g/cm}^3\). In this scenario the several objects have been discovered by the authors (see Refs. [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21]). It has also been investigated that the pressure anisotropy can disturb the critical mass, compactness, and steadiness of the extremely dense relativistic compact objects. Recently Chaichian et al. [22], Ferrer et al. [23] have proposed that the pressure anisotropy can be produced by the action of the magnetic field on a Fermi gas. Recently, Maurya et al. [24, 25, 26, 27, 28, 29] have discovered several charged and uncharged solution for the compact stars. The effect and role of anisotropy with and without equation of state (EOS) were proposed by Harko and Mak [30], Maurya et al. [31, 32, 33, 34, 35], Varela et al. [36]. On the other hand, Sharma and Maharaj [37] have obtained an analytical extended solution by taking a specific choice of mass function. In the current days, the embedding class one solution has much attraction towards the researchers. Many mathematicians and physicists have explained the concept of embedding that the curved spacetime can be embedded into higher dimensional flat spacetime. The Riemann proposed the innovative idea of Riemannian geometry by giving an exhaustive study of the geometric objects in 1850. Sudden after, some mathematicians proposed some concept that the Riemannian manifold can be embedded into Euclidian space of higher dimensional. This concept was known as the isometric embedding problem which was unproved for many years. The first time in 1871, the Schlaefli guessed the prospect of embedding the Riemannian manifold locally and isometrically into the higher-dimensional Euclidean space under some constraint. Later on this conjecture was proved and founded by Janet [38], Cartan [39] and Burstin [40]. They have proved that any l dimensional Riemannian space can always be locally embedded into any pseudo-Euclidean space of dimension \(L\ge l(l+1)/2\). In the case of symmetric, we need at least \(L=(l+1)\)-dimension of pseudo-Euclidian flat space for the embedding of l dimensional Riemannian geometry isometrically. This embedding theory has been also well explained by Stephani [41] and his collaborators in their pioneering work. In this concerns, the most popular work on embedding class one have been done by several authors [42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55].
The article is organised as follows: In Sect. 2, we defined the field equations for the charged matter distribution via gravitational decoupling (GD) and corresponding interior space-time. For this purpose, we divide this Sect. 2 into three Sects. 2.1, 2.2 and 2.3. In Sects. 2.1 and 2.2, we defined the action for two sources and corresponding field equations for both sources, respectively. The embedding class one condition is given in Sect. 2.3. We have obtained new solution for charged anisotropic matter distribution via GD in Sect. 3. In Sect. 4, we have performed the matching condition to determined the arbitrary constants and necessary parameters. The physical analysis has been presented in Sect. 4 which consist three subsections namely Sect. 5.1. Regularity of gravitational decoupling model, Sect. 5.2. Equilibrium condition for GD model, and Sect. 5.3. Stability of the stellar model via cracking. The mass-radius ratio and surface redshift are given in Sect. 6 where we discussed the gravitational and effective mass. The final section has been made for discussion and conclusion.
2 The field equations for the charged matter distribution via gravitational decoupling (GD) and corresponding interior space-time
2.1 The action for two sources
2.2 The field equations for \(T_{\mu \nu }\) and \(\varTheta _{\mu \nu }\) via MGD approach
2.3 Embedding class one condition associated with the geometry {\(T_{\mu \nu }\), \(\xi \), \(\mu \)}
3 New embedding class one solution by gravitational decoupling
4 Matching conditions
5 Physical analysis
5.1 Regularity of the gravitational decoupling model
5.2 Equilibrium condition for GD model
These above forces form the equilibrium of the system. The variation of these forces is intensely connected to the constant parameter \(\alpha \). Figure 4 shows that how we reached the equilibrium of the GD system under these different forces. Now we will focus on several points based on the different values taken for establishing the equilibrium stage. We have plotted the Fig. 4 for several values of \(\alpha \). The top left figure is plotted of \(\alpha =0.0\). As we see that the electric force (\(F_e\)) and extra coupling force (\(F_{\alpha }\)) have no effect to balance this mechanism however the gravitational force (\(F_g\)) is balanced by joint action of hydrostatic (\(F_h\)) and anisotropic (\(F_a\)) forces. On the other hand, the Figures top right, bottom left and bottom right have been plotted for \(\alpha =0.10\), \(\alpha =0.25\) and \(\alpha =0.50\), respectively. From these figures, it is interesting to note that the electric force \(F_{e}\) and coupling force \(F_{\alpha }\) have same values in magnitude within the star but they are acting in the opposite directions while \(F_g\) can be balanced by combining of \(F_a\) and \(F_h\). Therefore, the equilibrium is achieved for each different value of \(\alpha \).
5.3 Stability of the stellar model via cracking
6 Mass-radius ratio and surface redshift
6.1 Gravitational and effective mass
6.2 Surface and gravitational redshift
The numerical values of physical parameters and constants of the compact star of mass \(M=1.04M_{\odot }\) and radius \(R=7.67\) km for different values of \(\alpha \)
\(\alpha \) | Central pressure, \((p)_c^{\text {eff}}\) | Central density, \(\rho _c^{\text {eff}}\) | Surface density, \(\rho _s^{\text {eff}}\) | Electric charge at surface (Q) | a | D | C |
---|---|---|---|---|---|---|---|
0.0 | \(1.37465\times 10^{35}\) | \(1.53516\times 10^{15}\) | \(8.76814\times 10^{14}\) | 0.0000 | 0.00155 | 6.16134 | 0.42351 |
0.10 | \(1.40961\times 10^{35}\) | \(1.51576\times 10^{15}\) | \(8.75847\times 10^{14}\) | \(0.5593\times 10^{20}\) | 0.00154 | 6.10721 | 0.41976 |
0.25 | \(1.46021\times 10^{35}\) | \(1.48819\times 10^{15}\) | \(8.74397\times 10^{14}\) | \(0.8748\times 10^{20}\) | 0.00153 | 6.02864 | 0.41418 |
0.35 | \(1.45184\times 10^{35}\) | \(1.48539\times 10^{15}\) | \(8.73592\times 10^{14}\) | \(1.0278\times 10^{20}\) | 0.00153 | 6.03438 | 0.41519 |
0.5 | \(1.53678\times 10^{35}\) | \(1.44435\times 10^{15}\) | \(8.71390\times 10^{14}\) | \(1.2156\times 10^{20}\) | 0.00152 | 5.90584 | 0.405315 |
Comparative study of lower bound, mass-radius ratio, upper bound, compactness \((u=M_{\text {eff}}/R)\) and surface red-shift of the star for different values of \(\alpha \)
\(\alpha \) | Lower bound, \(\frac{Q^2\,(18R^2+Q^2)}{2R^2\,(12R^2+Q^2)}\) | Mass-radius ratio (\(\frac{M}{R}\)) | Compactness, \( \frac{M_{\text {eff}}}{R}\) | Upper bound \(\frac{4R^2+3Q^2+2R\sqrt{R^2+3Q^2}}{9R^2}\) | Surface redshift, \(z_s\) |
---|---|---|---|---|---|
0.0 | 0.00000 | 0.20 | 0.20000 | 0.666666 | 0.29110 |
0.10 | 0.002935 | 0.20 | 0.198081 | 0.669272 | 0.286885 |
0.25 | 0.007178 | 0.20 | 0.195263 | 0.673026 | 0.280921 |
0.35 | 0.009910 | 0.20 | 0.1934616 | 0.6754363 | 0.277152 |
0.50 | 0.013853 | 0.20 | 0.190770 | 0.6789036 | 0.271582 |
7 Discussion and conclusions
Our study is dedicated to a new class of completely deformed solution for the spherically symmetric charged anisotropic compact stellar model. This completely deformed solution has been obtained by using the gravitational decoupling approach in the framework of embedding class one spacetime. In our investigations, we confirm that the ‘gravitational decoupling’ technique is a powerful mechanism to determine a more general solution of the Einstein field equations in the scenario of embedding class one spacetime. To be more specific, in this study first we define the action for the modified matter distribution via introducing another extra source. Then we write the general equation of motion by varying this action along with the metric tensor \(g^{\mu \nu }\). Next, we obtain the Einstein field equations of spherically symmetric metric corresponding to the above general equation of motion for effective energy-momentum tensor \(T^{\text {eff}}_{\mu \nu }\), which is the combination of two sources namely \(T_{\mu \nu }\) and \(\varTheta _{\mu \nu }\). By using gravitational decoupling \({{\xi }}~\mapsto ~ \nu = \xi +\alpha \ h(r)\) and \(e^{-{\mu }}~ \mapsto ~ e^{-\lambda } = e^{-{\mu }}+\alpha \,f(r)\), we separated the effective system into two separate systems as first corresponds to \(T_{\mu \nu }\) (known as Einstein field equations) and second for \(\varTheta _{\mu \nu }\) (Known as quasi-Einstein field equations). Now our aim is to solve both systems individually. As we see that the first system [Eqs. (33)–(35)] depend on two unknowns \(\xi \) and \(\mu \). To solve this system we chose a particular form of gravitational potential, \(\xi =C(1+ar^2)^4\), same as Durgapal IV solution which satisfies all the physical and mathematical requirements for a regular and well-behaved solution [89]. Then we apply the embedding class one condition to determine the solution for this system for anisotropic matter distribution. Now we will focus on second system which is given by the Eqs. (18)–(20). Before solving these equations, first we converted the quasi-Einstein Eqs. (18)–(20) into a Maxwell field Eqs. (36)–(38) by applying the Ovalle [58] approach. Now we see that field Eqs. (36)–(38) depend on mainly three unknowns as electric charge q, deformation functions h(r) and f(r). For determining the solution of this Maxwell system, we need to solve the conservation Eq. (40) which is a first-order ordinary differential (ODE) equation containing two unknown functions \(\varPsi \) and h(r), where (\(\varPsi ={q^2}/{r^4}\)). Therefore, we solve this ODE by taking a specific form of deformation function h(r) [which is non-singular at the center and increasing away from the centre, see Eqs. (50) and (53)], and then obtained a closed-form expression for electric field (Eq. 51). Then after we obtain an another deformation function f(r) [see Eq. (54)] by solving of the Eq. (36) together with Eq. (51). In this way, we have solved both systems completely and the obtained solution for this complete system is given by line element (57). Moreover, we also determined the effective quantities density, pressures (radial and tangential) and anisotropy factor to describe the structure of the charged compact object. To perform the graphical analysis we used the matching conditions and obtain all the arbitrary constants. The physical analysis of the solution for the charged compact object is given as follows:
The behavior of electric charge q is shown in Fig. 1 for different values of \(\alpha \) with respect to specific mass-radius ratio \(M/R=0.2\). From this figure we see that electric charge is zero at centre and increasing towards the boundary of star. We have also calculated the amount of charge on the boundary in Coulomb unit as: (i) \(Q=0.5593\times 10^{20} \text {C}\) for \(\alpha =0.10\), (ii) \(Q=0.8748\times 10^{20} \text {C}\) for \(\alpha =0.25\), (iii) \(Q=1.0278\times 10^{20} \text {C}\) for \(\alpha =0.35\), (iv) \(Q=1.2156\times 10^{20} \text {C}\) for \(\alpha =0.50\). This amount of charge shows that it is increasing as \(\alpha \) increases.
In Fig. 2, the variation of deformations function h(r) and f(r) are given by top (left and right) figures while the bottom (left and right) figures are corresponding to gravitational potentials, viz. \(e^{\lambda }\) and \(e^{\nu }\) for different \(\alpha \) and fixed \(M/R=0.2\). It can be clearly seen that for each different values of \(\alpha \), the both deformation functions and gravitational potential are free from singularity and increasing throughout within the compact stellar model. The such behavior of h(r), f(r), \(e^{\lambda }\) and \(e^{\nu }\) gives a physically well-behaved stellar model. The variation of effective matter distribution \(p^{\text {eff}}_r\) and \(p^{\text {eff}}_t\) against the radial coordinate r / R are given shown in top (left and right) of Fig. 3 while \(\rho ^{\text {eff}}\) and \(\varDelta ^{\text {eff}}\) are featured in bottom (left and right) of this Fig. 3. From this figure, we observe that effective pressures and density are maximum at the centre and decreasing outward. Also, the effective anisotropy is zero at the centre (i.e. radial and tangential pressure are equal at center) and positive throughout within in stellar model. This feature of anisotropy shows that the force due to anisotropy is directed outward which allows the construction of a more compact object. Moreover, the numerical values for central pressure, central and surface density are given in Table 1. The surface density of the present model lies between \(3.131\rho _s-3.112\rho _s\) which is higher than the normal nuclear density \(\rho _s\) [9, 90, 91]. The above range of surface density confirm that the obtained stellar structure is an appropriate candidate for the hypothetical ultradense compact object.
To check the equilibrium position of all forces, we have studied the generalized TOV equation for a gravitationally decoupled system. In Fig. 4, we have shown the distributions of all forces \(F_h\), \(F_a\), \(F_g\), \(F_e\) and \(F_{\alpha }\) for the coupling parameter \(\alpha =0.0\) (top left), \(\alpha =0.10\) (top right), \(\alpha =0.25\) (bottom left) and \(\alpha =0.50\) (bottom right). From these individual figures, we see that all forces achieve the equilibrium condition, which shows the stability of the system. In this gravitational decoupled system, a new kind of force \(F_{\alpha }\) is coming due to the geometric deformation in the gravitational potential source. For all values of \(\alpha >0\), this extra force \(F_{\alpha }\) acts along inward direction and behaves an attractive nature which is balanced by electric force \(F_e\) such that \(F_e +F_\alpha =0\), this yields an equilibrium condition for the extra source \(\varTheta _{\mu \nu }\). On the other hand, the gravitational force \(F_{g}\) is balanced by the joint action of hydrostatic force \(F_h\) and anisotropic force \(F_a\) such that \(F_g+F_h+F_a=0\), which provides the stable equilibrium for the matter distribution \(T_{\mu \nu }\). By combining of both equilibrium conditions, we get \(F_g+F_h+F_a+F_e +F_\alpha =0\). This last equation shows that our complete system is in equilibrium stage.
The causality and stability analysis of the model have been shown in Figs. 5 and 6, respectively. From Fig. 5, we observe that both velocities are satisfying the causality condition, i.e. \(0<\text {v}^2_r<1\) and \(0<\text {v}^2_t<1\), everywhere within the star. Moreover, it is interesting to see from this figure that the radial speed of sound (\(\text {v}^2_r\)) is decreasing throughout for \(0<\alpha <0.27\) while if \(\alpha \ge 0.27\), then \(\text {v}^2_r\) decreases first and reach the minimum level and then start increasing towards the boundary. This feature of the \(\text {v}^2_r\) shows that the solution is not well behaved for higher values of \(\alpha \) (Lake and Delgaty [89]). On the other hand, the tangential speed of sound is decreasing throughout the compact object for all values of \(\alpha \). Now we will focus on the stability of the model via Aberu criterion (known as Herrera cracking concept). For this purpose, we plotted the stability factors \(\text {v}^2_t-\text {v}^2_r\) (upper panel) and \(\text {v}^2_r-\text {v}^2_t\) (lower panel) against the radial coordinate r / R in Fig. 6. From this figure it can be clearly observed that both factors lie within the stability range i.e. \(0<\text {v}^2_r-\text {v}^2_t<1\) and \(-1<\text {v}^2_t-\text {v}^2_r<0\). This stability range shows that the radial velocity is always greater than the tangential velocity throughout the star. Therefore, the obtained stellar model is stable.
Now at last we discuss the most important physical features as gravitational and effective mass, and surface redshift of the present model. It is argued that the gravitational mass in always greater than effective mass in presence of electric charge within the matter distribution. However, both masses are same for the perfect or anisotropic fluid matter distribution i.e. only in absence of charge (see subsection 6.1 for complete details). In Table 2, the numerical values of the effective mass-radius ratio \(\big (\frac{M_{\text {eff}}}{R}\big )\) are given for different \(\alpha \) with fixed gravitational mass-radius ratio \(\big (\frac{M}{R}\big )\). From this table we have also noticed that the ratio \(\frac{M_{\text {eff}}}{R}\) is decreasing for increasing \(\alpha \). Since the surface redshift depends on the compactness \(\big (u=\frac{M_{\text {eff}}}{R}\big )\) then the surface redshift will also decrease if \(\alpha \) increases which can be seen in Table 2. The obtained surface redshift for different values of \(\alpha \) is given as follows: (i) \(\text {z}_s=0.2911\) for \(\alpha =0.0\), (ii) \(\text {z}_s=0.286885\) for \(\alpha =0.10\), (iii) \(\text {z}_s=0.280921\) for \(\alpha =0.25\), (iv) \(\text {z}_s=0.277152\) for \(\alpha =0.35\), and (v) \(\text {z}_s=0.271582\) for \(\alpha =0.50\). The Fig. 7 shows the behavior gravitational redshift inside the charged compact star, which is maximum at centre and decreasing towards the surface boundary, and attains minimum at surface. The above obtained surface redshift is compatible with the values proposed by Ivanov [2] and Bowers and Liang [8]. Finally, we would like to mention that the geometric deformation gravitational decoupling approach is a very effective technique to solve the Einstein field equations for a more complex system.
Notes
Acknowledgements
S.K. Maurya acknowledges to the administration of University of Nizwa for their continuous support and encouragement.
References
- 1.J. Lattimer, J. Prakash, Phys. Rev. Lett. 94, 111101 (2005)ADSCrossRefGoogle Scholar
- 2.B.V. Ivanov, Phys. Rev. D 65, 104011 (2002)ADSCrossRefGoogle Scholar
- 3.N.K. Glendenning, C. Kettner, F. Weber, Phys. Rev. Lett. 74, 3519 (1995)ADSCrossRefGoogle Scholar
- 4.S.L. Shapiro, S.A. Teukolosky, Black Holes, White Dwarfs and Neutron Stars: The Physics of Compact Objects (Wiley, New York, 1983)CrossRefGoogle Scholar
- 5.E. Witten, Phys. Rev. D 30, 272 (1984)ADSCrossRefGoogle Scholar
- 6.L. Herrera, W. Barreto, Phys. Rev. D 88, 084022 (2013)ADSCrossRefGoogle Scholar
- 7.X.Y. Lai, R.X. Xu, Astropart. Phys. 31, 128 (2009)ADSCrossRefGoogle Scholar
- 8.R.L. Bowers, E.P.T. Liang, Astrophys. J. 188, 657 (1974)ADSCrossRefGoogle Scholar
- 9.R. Ruderman, Rev. Astron. Astrophys. 10, 427 (1972)ADSCrossRefGoogle Scholar
- 10.M.K. Mak, T. Harko, Proc. R. Soc. A 459, 393 (2003)ADSCrossRefGoogle Scholar
- 11.L. Herrera, N.O. Santos, Phys. Rep. 286, 53 (1997)ADSMathSciNetCrossRefGoogle Scholar
- 12.L. Herrera, J. Ospino, A.D. Prisco, Phys. Rev. D 77, 027502 (2008)ADSMathSciNetCrossRefGoogle Scholar
- 13.L. Herrera, J. Ponce de Leon, J. Math. Phys. 26, 2302 (1985)ADSMathSciNetCrossRefGoogle Scholar
- 14.K. Dev, M. Gleiser, Gen. Relativ. Gravit. 34, 1793 (2002)CrossRefGoogle Scholar
- 15.K. Dev, M. Gleiser, Gen. Relativ. Gravit. 35, 1435 (2003)ADSCrossRefGoogle Scholar
- 16.M.H. Murad, S. Fatema, Eur. Phys. J. C 75, 533 (2015)ADSCrossRefGoogle Scholar
- 17.B.V. Ivanov, Eur. Phys. J. C 77, 738 (2017)ADSCrossRefGoogle Scholar
- 18.M. Jasim, D. Deb, S. Ray, Y.K. Gupta, S.R. Chowdhury, Eur. Phys. J. C 78, 603 (2018)ADSCrossRefGoogle Scholar
- 19.D. Deb, M. Khlopov, F. Rahaman, S. Ray, B.K. Guha, Eur. Phys. J. C 78, 465 (2018)ADSCrossRefGoogle Scholar
- 20.S.K. Maurya, S.D. Maharaj, D. Deb, Eur. Phys. J. C 79, 170 (2019)ADSCrossRefGoogle Scholar
- 21.P. Mafa Takisa, S.D. Maharaj, A.M. Manjonjo, S. Moopanar, Eur. Phys. J. C 77, 713 (2017)ADSCrossRefGoogle Scholar
- 22.M. Chaichian et al., Phys. Rev. Lett. 84, 5261 (2000)ADSCrossRefGoogle Scholar
- 23.E.J. Ferrer et al., Phys. Rev. C 82, 065802 (2010)ADSCrossRefGoogle Scholar
- 24.S.K. Maurya, Y.K. Gupta, Pratibha, Int. J. Mod. Phys. D 20, 1289 (2011)ADSCrossRefGoogle Scholar
- 25.N. Pant, S.K. Maurya, Appl. Math. Comput. 218(17), 8260–8268 (2012)MathSciNetGoogle Scholar
- 26.S.K. Maurya, Y.K. Gupta, Nonlinear Anal. Real World Appl. 13, 677 (2012)MathSciNetCrossRefGoogle Scholar
- 27.S.K. Maurya, Y.K. Gupta, S. Ray, V. Chatterjee, Astrophys. Space Sci. 361(10), 351 (2016)ADSCrossRefGoogle Scholar
- 28.S.K. Maurya, Y.K. Gupta, S. Ray, Eur. Phys. J. C 77, 360 (2017)ADSCrossRefGoogle Scholar
- 29.S.K. Maurya, S. Ray, A. Aziz, M. Khlopov, P. Chardonnet, Int. J. Mod. Phys. D 28, 1950053 (2019)ADSCrossRefGoogle Scholar
- 30.T. Harko, M.K. Mak, Annalen der Physik 11, 3 (2002)ADSCrossRefGoogle Scholar
- 31.S.K. Maurya, Y.K. Gupta, B. Dayanandan, M.K. Jasim, A. AlJamel, Int. J. Mod. Phys. D 26, 1750002 (2017)ADSCrossRefGoogle Scholar
- 32.S.K. Maurya, Y.K. Gupta, S. Ray, B. Dayanandan, Eur. Phys. J. C 75, 225 (2015)ADSCrossRefGoogle Scholar
- 33.S.K. Maurya, A. Banerjee, S. Hansraj, Phys. Rev. D 97, 044022 (2018)ADSMathSciNetCrossRefGoogle Scholar
- 34.S.K. Maurya, S.D. Maharaj, Jitendra Kumar, Amit Kumar Prasad, Gen. Relativ. Gravit. 51, 86 (2019)ADSCrossRefGoogle Scholar
- 35.S.K. Maurya, A. Banerjee, M.K. Jasim, J. Kumar, A.K. Prasad, A. Pradhan, Phys. Rev. D 99, 044029 (2019)ADSMathSciNetCrossRefGoogle Scholar
- 36.V. Varela, F. Rahaman, S. Ray, K. Chakraborty, M. Kalam, Phys. Rev. D 82, 044052 (2010)ADSCrossRefGoogle Scholar
- 37.R. Sharma, S.D. Maharaj, Mon. Not. R. Astron. Soc. 375, 1265 (2007)ADSCrossRefGoogle Scholar
- 38.M. Janet, Ann. Soc. Math. Polon. 5, 38 (1926)Google Scholar
- 39.E. Cartan, Ann. Soc. Math. Polon. 6, 1 (1927)Google Scholar
- 40.C. Burstin, Mat. Sb. 38, 74 (1931)Google Scholar
- 41.H. Stephani, D. Kramer, M.A.H. MacCallum, C. Hoenselaers, E. Herlt, Exact Solutions of Einsteins Field Equations (Cambridge University Press, Cambridge, 2003)zbMATHCrossRefGoogle Scholar
- 42.S.K. Maurya, Y.K. Gupta, B. Dayanandan, S. Ray, Eur. Phys. J. C 76, 266 (2016)ADSCrossRefGoogle Scholar
- 43.S.K. Maurya, Y.K. Gupta, T.T. Smitha, F. Rahaman, Eur. Phys. J. A 52, 191 (2016)ADSCrossRefGoogle Scholar
- 44.S.K. Maurya, S.D. Maharaj, Eur. Phys. J. A 54, 68 (2018)ADSCrossRefGoogle Scholar
- 45.S.K. Maurya, S.D. Maharaj, Eur. Phys. J. C 77, 328 (2017)ADSCrossRefGoogle Scholar
- 46.P. Bhar, K.N. Singh, N. Sarkar, F. Rahaman, Eur. Phys. J. C 77, 596 (2017)ADSCrossRefGoogle Scholar
- 47.S.K. Maurya, M. Govender, Eur. Phys. J. C 77, 347 (2017)ADSCrossRefGoogle Scholar
- 48.K.N. Singh, N. Pant, M. Govender, Eur. Phys. J. C 77, 100 (2017)ADSCrossRefGoogle Scholar
- 49.S. Hansraj, Eur. Phys. J. C 77, 557 (2017)ADSCrossRefGoogle Scholar
- 50.P. Bhar, S.K. Maurya, Y.K. Gupta, T. Manna, Eur. Phys. J. A 52, 312 (2016)ADSCrossRefGoogle Scholar
- 51.S.K. Maurya, Y.K. Gupta, S. Ray, D. Deb, Eur. Phys. J. C 77, 45 (2017)ADSCrossRefGoogle Scholar
- 52.K.N. Singh, N. Pant, N. Pradhan, Astrophys. Space Sci. 361, 173 (2016)ADSCrossRefGoogle Scholar
- 53.K.N. Singh, N. Pant, Astrophys. Space Sci. 361, 177 (2016)ADSCrossRefGoogle Scholar
- 54.N. Pant, K.N. Singh, N. Pradhan, Indian J. Phys. 91, 343 (2017)ADSCrossRefGoogle Scholar
- 55.K.N. Singh, P. Bhar, N. Pant, Int. J. Mod. Phys. D 25, 1650099 (2016)ADSCrossRefGoogle Scholar
- 56.J. Ovalle, Phys. Rev. D 95, 104019 (2017)ADSMathSciNetCrossRefGoogle Scholar
- 57.R. Casadio, J. Ovalle, R. da Rocha, Class. Quantum Gravity 32, 215020 (2015)ADSCrossRefGoogle Scholar
- 58.J. Ovalle, Phys. Lett. B 788, 213 (2019)ADSMathSciNetCrossRefGoogle Scholar
- 59.J. Ovalle, Mod. Phys. Lett. A 23, 3247 (2008)ADSMathSciNetCrossRefGoogle Scholar
- 60.J. Ovalle, Braneworld stars: anisotropy minimally projected onto the brane, in Gravitation and Astrophysics (ICGA9), ed. by J. Luo (World Scientific, Singapore, 2010)Google Scholar
- 61.J. Ovalle, F. Linares, Phys. Rev. D 88, 104026 (2013)ADSCrossRefGoogle Scholar
- 62.J. Ovalle, R. Casadio, R. da Rocha, A. Sotomayor, Eur. Phys. J. C 78, 122 (2018)ADSCrossRefGoogle Scholar
- 63.J. Ovalle, R. Casadio, R. da Rocha, A. Sotomayor, Z. Stuchlik, EPL 124, 20004 (2018)ADSCrossRefGoogle Scholar
- 64.E. Contreras, A. Rincon, P. Bargueno, Eur. Phys. J. C 79, 216 (2019)ADSCrossRefGoogle Scholar
- 65.J. Ovalle, C. Posada, Z. Stuchlík, Class. Quantum Gravity 36, 205010 (2019)CrossRefGoogle Scholar
- 66.L. Gabbanelli, J. Ovalle, A. Sotomayor, Z. Stuchlik, R. Casadio, Eur. Phys. J. C 79, 486 (2019)ADSCrossRefGoogle Scholar
- 67.R. Casadio, E. Contreras, J. Ovalle, A. Sotomay, Z. Stuchlik, Eur. Phys. J. C 79, 826 (2019)ADSCrossRefGoogle Scholar
- 68.E. Contreras, Eur. Phys. J. C 78, 678 (2018)ADSCrossRefGoogle Scholar
- 69.M. Sharif, S. Sadiq, Eur. Phys. J. C 78, 410 (2018)ADSCrossRefGoogle Scholar
- 70.E. Contreras, P. Bargueno, Eur. Phys. J. C 78, 558 (2018)ADSCrossRefGoogle Scholar
- 71.C. Las Heras, P. Leon, Fortsch. Phys. 66, 070036 (2018)ADSCrossRefGoogle Scholar
- 72.E. Morales, F. Tello-Ortiz, Eur. Phys. J. C 78, 618 (2018)ADSCrossRefGoogle Scholar
- 73.G. Panotopoulos, A. Rincon, Eur. Phys. J. C 78, 851 (2018)ADSCrossRefGoogle Scholar
- 74.S.K. Maurya, F. Tello-Ortiz, Eur. Phys. J. C 79, 85 (2019)ADSCrossRefGoogle Scholar
- 75.E. Contreras, A. Rincon, P. Bargueno, Eur. Phys. J. C 79, 216 (2019)ADSCrossRefGoogle Scholar
- 76.E. Contreras, Class. Quantum Gravity 36, 095004 (2019)ADSCrossRefGoogle Scholar
- 77.K.N. Singh, S.K. Maurya, M.K. Jasim, F. Rahaman, Eur. Phys. J. C 79, 851 (2019)ADSCrossRefGoogle Scholar
- 78.K.R. Karmarkar, Proc. Indian Acad. Sci. A 27, 56 (1948)CrossRefGoogle Scholar
- 79.S.N. Pandey, S.P. Sharma, Gen. Relativ. Gravit. 14, 113 (1981)ADSCrossRefGoogle Scholar
- 80.K. Lake, Phys. Rev. D 67, 104015 (2003)ADSMathSciNetCrossRefGoogle Scholar
- 81.M.C. Durgapal, R.S. Fuloria, Gen. Relativ. Gravit. 17, 671 (1985)ADSCrossRefGoogle Scholar
- 82.W. Israel, Nuovo Cim. B 44, 1 (1966)ADSCrossRefGoogle Scholar
- 83.G. Darmois, Mémorial des Sciences Mathematiques (Gauthier-Villars, Paris, 1927), Fasc. 25 (1927)Google Scholar
- 84.H. Abreu, H. Hernández, L.A. Núñez, Calss. Quantum Gravity 24, 4631 (2007)ADSCrossRefGoogle Scholar
- 85.L. Herrera, Phys. Lett. A 165, 206 (1992)ADSCrossRefGoogle Scholar
- 86.H.A. Buchdahl, Phys. Rev. D 116, 1027 (1959)ADSCrossRefGoogle Scholar
- 87.H. Andreasson, J. Differ. Equ. 245, 2243 (2008)ADSCrossRefGoogle Scholar
- 88.C.G. Bohmer, T. Harko, Gen. Relativ. Gravit. 39, 757 (2007)ADSCrossRefGoogle Scholar
- 89.M.S.R. Delgaty, K. Lake, Comput. Phys. Commun. 115, 395 (1998)ADSCrossRefGoogle Scholar
- 90.N.K. Glendenning, Compact Stars: Nuclear Physics, Particle Physics and General Relativity (Springer, New York, 1997)zbMATHGoogle Scholar
- 91.M. Herzog, F.K. Ropke, Phys. Rev. D 84, 083002 (2011)ADSCrossRefGoogle Scholar
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