Some new models of strange stars in 5-D Einstein-Gauss-Bonnet Gravity

Abstract: In this paper, we present some new models for anisotropic compact stars within the framework of 5-dimensional Einstein-Gauss-Bonnet (EGB) gravity with a linear and nonlinear equation of state considering a metric potential proposed for Thirukkanesh and Ragel (2012) and generalized for Malaver (2014). The new obtained models satisfy all physical requirements of a physically reasonable stellar object. Variables as energy density, radial pressure and the anisotropy are dependent of the values of the Gauss-Bonnet coupling constant.


Introduction
Mathematical modeling within the framework of the general theory of relativity has been used to explain the behavior and structure of massive objects as neutron stars, quasars, black holes, pulsars and white dwarfs [1,2] and requires finding the exact solutions of the Einstein-Maxwell system [3]. A detailed and systematic analysis was carried out by Delgaty and Lake [4] which obtained several analytical solutions that can describe realistic stellar configurations.
It is very important to mention the pioneering works of Schwarzschild [5], Tolman [6], Oppenheimer and Volkoff [7] and Chandrasekhar [8] in the development of the first theoretical models of stellar objects. Schwarzschild [5] obtained interior solutions that allows describing a star with uniform density, Tolman [6] generated new solutions for static spheres of fluid, Oppenheimer and Volkoff [7] studied the gravitational equilibrium of neutron masses using the equation of state for a cold Fermi gas and general relativity and Chandrasekhar [8] produced new models of white dwarfs in presence of relativistic effects. Some of these results have been extended to higher dimensions and the dimensionality of space-time apparently influence the stability of these fluid spheres [9].
Recently, astronomical observations of compact objects have allowed new findings of neutron stars and strange stars that adjust to the exact solutions of the 4-D Einstein field equations and the data on mass maximum, redshift and luminosity are some of the most relevant characteristics for verifying the physical requirements of these models [10]. A great number of exact models from the Einstein-Maxwell field equations have been generated by Gupta and Maurya [11], Kiess [12], Mafa Takisa and Maharaj [13], Malaver and Kasmaei [14], Malaver [15,16], Ivanov [17] and Sunzu et al [18]. In the development of these models, several forms of equations of state can be considered [19]. Komathiraj and Maharaj [20], Malaver [21], Bombaci [22], Thirukkanesh and Maharaj [23], Dey et al. [24] and Usov [25] assume linear equation of state for quark stars. Feroze and Siddiqui [26] considered a quadratic equation of state for the matter distribution and specified particular forms for the gravitational potential and electric field intensity. MafaTakisa and Maharaj [13] obtained new exact solutions to the Einstein-Maxwell system of equations with a polytropic equation of state. Thirukkanesh and Ragel [27] have obtained particular models of anisotropic fluids with polytropic equation of state which are consistent with the reported experimental observations. Malaver [28] generated new exact solutions to the Einstein-Maxwell system considering Van der Waals modified equation of state with polytropic exponent. Tello-Ortiz et al. [29] found an anisotropic fluid sphere solution of the Einstein-Maxwell field equations with a modified version of the Chaplygin equation of state.
The analysis of compact objects with anisotropic matter distribution is very important, because that the anisotropy plays a significant role in the studies of relativistic spheres of fluid [30][31][32][33][34][35][36][37][38][39][40][41][42]. Anisotropy is defined as Δ = − where is the radial pressure and is the tangential pressure. The existence of solid core, presence of type 3A superfluid [43], magnetic field, phase transitions, a pion condensation and electric field [25] are most important reasonable facts that explain the presence of tangential pressures within a star. Many astrophysical objects as X-ray pulsar, Her X-1, 4U1820-30 and SAXJ1804.4-3658 have anisotropic pressures. Bowers and Liang [42] include in the equation of hydrostatic equilibrium the case of local anisotropy. Bhar et al. [44] have studied the behavior of relativistic objects with locally anisotropic matter distribution considering the Tolman VII form for the gravitational potential with a linear relation between the energy density and the radial pressure. Malaver [45][46], Feroze and Siddiqui [26,47] and Sunzu et al. [18] obtained solutions of the Einstein-Maxwell field equations for charged spherically symmetric space-time by assuming anisotropic pressure.
The behavior and dynamics of the gravitational field can be extended to higher dimensions [48]. The history of higher dimensions goes back to the work done by Kaluza [49] and Klein [50] who introduced the concept of extra dimensions in addition to the usual four dimensions (4-D) to unify gravitational and electromagnetic interactions. In general theory of relativity, the results obtained in four dimensions can be generalized in higher dimensional context and study the effects due to incorporation of extra space-time dimensions [51]. Within this framework, a very useful and fruitful generalization is the Einstein-Gauss-Bonnet gravity, which has generated a lot of interest among researchers and has been influenced by many scientists working in this field [52]. The modeling of compact objects in EGB gravity has shown that some physical variables are modified when they are compared to their 4-D counterparts, but the condition of the Schwarzschild constant density sphere has been demonstrated in EGB gravity [10]. Recently, Bhar et al. [53] performed a comparative study of compact objects in five dimensions (5-D) between EGB gravity and classical general relativity theory and found that many features as stability, causality and energy conditions remain unaffected in the stellar interior.
In this work, we have used the Thirukkanesh-Ragel-Malaver ansatz [27,37,54] in order to generate some stellar models with anisotropic matter distribution in EGB gravity. The system of field equations has been solved to obtain analytic solutions which are physically acceptable. The paper is organized as follows: In Section.2, we present the framework of EGB gravity. The modified Einstein-Maxwell field equations with the Gauss-Bonnet coupling constant are presented in Section.3. With the Thirukkanesh-Ragel-Malaver ansatz, we generate some models of an anisotropic star with a linear and nonlinear equation of state within EGB gravity in Section.4. In Section. 5, physical requirements for the new models are described. In Section.6, a physical analysis of the new solutions is performed. In final Section, we conclude.

Einstein-Gauss-Bonnet Gravity
The Gauss-Bonnet action in five dimensions can be written as where α is the Gauss-Bonnet coupling constant. The strength of the action LGB lies in the fact that despite the Lagrangian being quadratic in the Ricci tensor, Ricci scalar and the Riemann tensor, the equations of motion turn out to be second order quasi-linear which are compatible with Einstein's theory of gravity [52,53]. The EGB field equations may be written as where represents the Einstein tensor, is the total energy-momentum tensor and the Lanczos tensor is given by where the Lovelock term has the form

Field Equations
The 5-dimensional line element for a static spherically symmetric space-time takes the form where the metric functions and are the gravitational potentials. By considering the commoving fluid velocity as = , the EGB field equations (2) reduce to Here primes means a derivation with respect to the radial coordinates r and ρ is the energy density, is the radial pressure and is the tangential pressure. With the transformations = , ( ) = 2λ and ( ) = 2ν suggested by Durgapal and Bannerji [55] and with c>0 as arbitrary constant, the field equations (6)-(8) can be written as follows where = 4 contains the Gauss-Bonnet coupling constant α and dots denote differentiation with respect to .
In this paper, we imposed the following equations of state, linear and quadratic, respectively, relating the radial pressure to the energy density, where γ is a positive constant = and = (12)

The New Anisotropic Models
In this research, we take the form of the gravitational potential Z(x) as = (1 − ) proposed for Thirukanesh and Ragel [27] and subsequently generalized by Malaver [37], taking as an arbitrary parameter. This potential is regular at the stellar center and well behaved in the interior of the sphere. Using Z(x) in equation (9), we obtain Substituting the equation (13) With Z(x) and (14) in equation (10), we have Integrating equation (15) with respect to , we obtain and is the constant of integration.
For the metric functions and ,we have and the anisotropy can be written as With the quadratic equation of state, we obtain for the radial pressure = (12 + 48 − 9(1 + 8 ) + 24 ) and for the equation (10),we have Integrating (24) with respect to , we obtain Again the constants D, E, F, G, H and I are given by For the anisotropy Δ ,we have

Physical Acceptability in EGB Gravity
For a model to be physically acceptable in EGB gravity, the following conditions should be satisfied [10,53]: (i) The metric potentials and assume finite values throughout the stellar interior and are singularity-free at the center r=0.
(ii) The energy density ρ and the radial pressure should be positive inside the star.
(iii) The anisotropy is zero at the center r=0, i.e. Δ(r=0) =0. (iv) The energy density and radial pressure are decreasing functions with the radial parameter, i.e.
(v) Any physically acceptable model must satisfy the causality condition, that is, for the radial sound speed = ,we should have 0 ≤ ≤ 1 .
where R is the radius of the star and In Equation. (36) ,M is associated with the gravitational mass of the hypersphere. .We show that in r=0 ,

Physical Features of the New Models
For this case, the causality condition 0 ≤ ≤ 1 implies that (46) Again, with the first fundamental form, we can obtain and for the second fundamental form From the equation (44) been considered the radius R= 5.7 Km and c=1. The Figures 1 and 2 present the dependency of ρ and with the radial coordinates, respectively.     In two cases, linear and quadratic, the energy density remains positive, continuous and is monotonically decreasing function throughout the stellar interior ( Figure 1). It is also noted that the density increases with increasing α. The radial variation of energy density gradient has been shown in Figure 2, in which it is observed that < 0 in EGB gravity.
In the linear regimen, the radial pressure showed the same behavior by the energy density, that is, it is growing within the star and vanishes at a greater radial distance, but takes the higher values when α is increased and its results are shown in Figure 3. Again ,according to Figure 4, the profile of shows that radial pressure gradient is negative inside the stellar interior. The anisotropic factor is plotted in Figure 5 and it shows that vanishes at the centre of the star, i.e. Δ(r=0) =0 [30]. We can also note that Δ admits lower values when α increases.
The Figures 6,7,8 and 9 show the dependence of , , and Δ respectively with the radial coordinates in the quadratic case for the different values of coupling constant α . In all the cases, it has been considered R= 5.7 Km, c=1and γ=1/3 .    As in the linear regimen, with the quadratic equation, the radial pressure always is positive inside the star and vanishes at a finite radial distance and its results are shown in Figure 6. Again, the radial pressure increases when α takes higher values. In the Figure 7, it is also verified that the gradient is negative in the stellar interior.
A physically acceptable model must satisfy the causality condition, i.e., the radial sound speed must be within the range 0 ≤ ≤ 1. The profile of radial speed sound is plotted in Figure 8 for different values of coupling constant α. In all the cases is in the expected range and is a monotonic decreasing function with the radial coordinates. Figure 9 shows that the anisotropy is zero at the center r=0 and its value increases towards the surface of the star. As in the linear case Δ takes lower values when α increases.

Conclusions
In this paper, we have generated new models of compact stars within the framework of Einstein-Gauss-Bonnet gravity. With the use of Thirukkanesh-Ragel-Malaver ansatz for the gravitational potentials and with a linear and quadratic equation of state, we are able to produce two classes of exact solutions of the EGB field equations. We show that the developed configuration obeys the rigorous conditions required for the physical viability of the stellar model. It is to be noted in EGB gravity that the coupling constant α has nonnegligible effects on the physical quantities such as energy density and radial pressure of the star which increases with an increase in α. As expected, the matching conditions require that the radial pressure vanishes at some finite radius of the stellar object and this defines the boundary of the star. An evidence to the effect of the coupling constant can be observed in the behavior of energy density. In the two studied cases, linear and quadratic, when the Gauss-Bonnet constant increases, the energy density also increases and it allows that these models can support more masses. It is also noted that for all the values of the coupling constant α, is maximum at the centre and it decreases radially. Also, it is observed that inside of the star 0 ≤ ≤ 1 , which shows that the models are stable. Within the framework of EGB gravity, it is plausible to consider that the proposed models can describe real compact objects such as white dwarfs, neutron stars and pulsars.