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Generalized Relativistic Anisotropic Compact Star Models by Gravitational Decoupling Eur. Phys. J. C (2019) 79:85 https://doi.org/10.1140/epjc/s10052-019-6602-1 Regular Article - Theoretical Physics Generalized relativistic anisotropic compact star models by gravitational decoupling S. K. Maurya1,a, Francisco Tello-Ortiz2,b 1 Department of Mathematics and Physical Science, College of Arts and Science, University of Nizwa, Nizwa, Sultanate of Oman 2 Departamento de Física, Facultad de ciencias básicas, Universidad de Antofagasta, Casilla 170, Antofagasta, Chile Received: 29 November 2018 / Accepted: 15 January 2019 / Published online: 29 January 2019 © The Author(s) 2019 Abstract In this work we have extended the Maurya-Gupta main ingredient of this scheme is to add an extra source to isotropic fluid solution to Einstein field equations to an aniso- the energy momentum tensor of a spherically symmetric and tropic domain. To do so, we have employed the gravitational static spacetime through a dimensionless constant β decoupling via the minimal geometric deformation approach. m = ˜ m + βθm, The present model is representing the strange star candidate Tn Tn n (1) LMC X-4. A mathematical, physical and graphical analy- ˜ m sis, shown that the obtained model fulfills all the criteria where Tn represents a perfect fluid matter distribution and θ m to be an admissible solution of the Einstein field equations. n is the extra source which encodes the anisotropy and Specifically, we have analyzed the regularity of the metric can be formed by scalar, vector or tensor fields. It is worth ˜ m potentials and the effective density, radial and tangential pres- mentioning that Tn does not necessarily should corresponds sures within the object, causality condition, energy condi- to a perfect fluid, in fact it can describes, for example, an tions, equilibrium via Tolman–Oppenheimer–Volkoff equa- anisotropic fluid, a charged fluid, etc. [18]. On the other tion and the stability of the model by means of the adiabatic hand it is well known that an astrophysical object not nec- index and the square of subliminal sound speeds. essarily meet the isotropic condition at all i.e pr = pt .Of course, a more realistic scenario has been shown that relax- ing the isotropic condition i.e pr = pt leads to a better 1 Introduction understanding of how compact structures like neutron stars, white dwarfs and black holes work. Theoretical researches Developed by Ovalle and collaborators [1–12], gravitational in the past developed by Ruderman [29] Canuto [30–32] decoupling by minimal geometric deformation (MGD here- and Canuto et al. [33–36] revealed that anisotropies can be inafter) approach, has proved to be a novel and powerful tool emerged when matter density is higher than the nuclear den- in order to obtain analytic solutions to Einstein field equations sity one. These pioneering works established a new branch described by an anisotropic fluid. Recently, several works within the study of the influence of local anisotropies in gen- available in the literature have proven the versatility of the eral relativity [37–68]. Furthermore, over the last decade a method, using it in different contexts such as extending solu- lot of works available in the literature have considering the tions of Einstein field equations described by perfect fluid to inclusion of anisotropic matter distribution in solving ana- anisotropic domains [13–17], Einstein–Maxwell equations lytical models to Einstein field equations [69–109](and ref- [18,19], black holes [20–22], cloud of strings [23], Einstein– erences contained therein). On the other hand as Mak and Klein–Gordon equations [24] and wi- thin the framework of Harko have argued [63], anisotropy can arise in different theories in higher dimensions [25]. Originally, the method contexts such as: the existence of a solid core or by the was developed to deform Schwarz- schild’s spacetime [1,2] presence of type 3A superfluid [110], pion condensation within the context of Randall-Sun- drum braneworld [26,27]. [111] or different kinds of phase transitions [112]. Of course However, it has been extended to another type of symmetry, the presence of anisotropy introduces several and novel fea- specifically spacetime with cylindrical symmetry [28]. The tures in the matter distribution, e.g. in presence of a positive anisotropy factor Δ ≡ pt − pr > 0, the stellar configuration a e-mail: [email protected] experiences a repulsive force (attractive in the case of nega- b e-mail: [email protected] tive anisotropy factor) that offsets the gravitational gradient. 123 85 Page 2 of 14 Eur. Phys. J. C (2019) 79 :85 Therefore, we obtain a more compact and massive objects in ter distribution is given as, distinction with the isotropic case [52,60,63]. Moreover, a m = ρ + ) m − δm + ( − )υmυ , positive anisotropy factor improves the stability and equilib- Tn pt u un pt n pr pt n (4) rium of the system. In the spirit of the above discussion we employ gravita- By using Eqs. (2) and (4) together, we get the following tional decoupling via MGD in order to extent the Maurya and equations Gupta [113] perfect fluid solution of Einstein field equations to an anisotropic scenario. To do so, we realize a geomet- −λ λ 1 1 e − + = 8πρ (5) ric deformation on the radial metric component encode by r r 2 r 2 ( ) a purely radial function g r and then we impose a suitable −λ ν 1 1 e + − = 8πp (6) election on it, according with the physical and mathemati- r r 2 r 2 r cal requirements, which leads us to know the contribution of −λ ν2 ν − λ νλ the extra source θ m to the material content. Of course, any e n ν + + − = 8πpt . (7) solution to Einstein field equations must meet some basics 2 2 r 2 criteria in order to be a well behaved and admissibly solution that can describes a realistic model [114]. So, the outline of The primes denote differentiation with respect to the radial the present work is as follows: Sect. 2 presents the mathemat- coordinate r. However, ρ, pr and pt represent the effective ical formulation of Einstein field equations for anisotropic density, the effective radial pressure and the effective tangen- matter distribution in the framework of gravitational decou- tial pressure respectively, which are given by pling via MGD, Sect. 3 is devoted to the construction of the ρ =˜ρ + βθt , anisotropic version of the Maurya and Gupta perfect fluid t (8) solution. In Sect. 4 we match the obtained model with the =˜− βθr , pr p r (9) exterior Schwarzschild spacetime in order to obtain the con- ϕ pt =˜p − βθϕ , (10) stant and physical parameters of the model, Sect. 5 analyzes all the basics criteria that an admissible solution to Einstein Here ρ˜ and p˜ represent the density and pressure for per- field equations must meet, such as regularity of the metric fect fluid distribution. We note that the presence of term θ potentials and the thermodynamic observables within the ϕ clearly introduces an anisotropy if θr = θϕ . Then the effec- stellar configuration, causality condition, energy conditions, r tive anisotropy is defined as, equilibrium and stability conditions. In Sect. 6, we discuss the behavior and form of effective metric potentials at very r ϕ Δ = p − p = β θ − θϕ . (11) large value of n i.e. n tends to infinity. Finally, Sect. 7 sum- t r r marizes and concludes the work. In order to solve Einstein’s equations we will introduce the gravitational decoupling via the MGD approach [12]. In this method, we transformed the metric potentials eν(r) and 2 Mathematical structure of Einstein equations for eλ(r) through a linear mapping which is given by, gravitational decoupling ν( ) ν(˜ ) e r −→ e r + β f (r) (12) Let us consider the curvature coordinates to describe the −λ(r) −→ −λ(˜ r) + β ( ) ultradense spherically symmetric stellar system, by using the e e g r (13) metric as follows [115], ν(˜ r) λ˜(r) Here e and e are gravitational metric potentials cor- ν( ) λ( ) f (r) ds2 = e r dt2 − e r dr2 − r 2 dθ 2 + sin2θdφ2 , (2) responding to perfect fluid matter distribution while and g(r) are the corresponding deformations functions. It’s worth mentioning that the foregoing deformations are purely ν( ) λ( ) where r and r are the metric potentials and function radial functions, this feature ensures the spherical symme- of the radial coordinate only. The energy-momentum tensor try of the solution. The so called MGD corresponds to set β associated with the dimensionless coupling constant can f (r) = 0org(r) = 0, in this case the deformation will be be read as, done only on the radial component, remaining the tempo- θ m m = ˜ m + βθm ral one unchanged. Then the anisotropic sector n is totally Tn Tn n (3) contained in the radial deformation (13). After replacing (13) ˜ m where the energy momentum tensor Tn corresponds to the into the system of Eqs. (5)–(7), it is decoupled in two systems perfect fluid solution for the metric (2) while the energy of equations. The first one corresponds to β = 0, it means m momentum tensor Tn corresponding to the anisotropic mat- perfect fluid matter distribution 123 Eur. Phys. J. C (2019) 79 :85 Page 3 of 14 85 ˜ (n−4−i) Ai (n−1)! ai −λ˜ λ 1 1 ai+1 − , ai+1 = − , A0 = e − + = 8πρ,˜ (14) n2−(i+2)n−(i+1) (i+1)! (n−i−2)! n+1 r r 2 r 2 a0 = 1 and An− j = an− j = 0ifn < j. However A, B and −2 C are arbitrary constants with dimension of C is length . ˜ ν˜ 1 1 −λ + − = π p˜ The metric (23) describes a static spherically symmetric e 2 2 8 (15) r r r configuration associated to an isotropic matter distribution −λ˜ 2 ˜ ˜ e ν˜ ν˜ − λ ν˜ λ which implies that radial and transverse pressures are equal ν˜ + + − = 8π p˜.
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