Eur. Phys. J. C (2019) 79:138 https://doi.org/10.1140/epjc/s10052-019-6642-6 Regular Article - Theoretical Physics Anisotropic compact star model: a brief study via embedding Piyali Bhara Department of Mathematics, Government General Degree College, Singur, Hooghly, West Bengal 712 409, India Received: 16 May 2018 / Accepted: 3 February 2019 / Published online: 14 February 2019 © The Author(s) 2019 Abstract In present article a new model of compact star is and other one is transverse pressure pt orthogonal to the for- = − 2 obtained in the framework of general relativity which does mer. pt pr is known as anisotropic factor and r not suffer from any kinds of singularity. We assume that the is anisotropic force which becomes attractive or repulsive underlying fluid distribution is anisotropic in nature along according to pr < pt or pr > pt . Bowers and Liang [3], with a new form for the metric potential eλ which is phys- in 1974, wrote a paper on the study of anisotropic distribu- ically reasonable. Though the model parameters depend on tion of matter which got wide attention. Anisotropy can be four constants a, b, A and B but we have shown that the revealed due to the existence of solid stellar core or by phase solutions depend on two free constants since these four con- transitions, pion condensation in a star [4], the presence of stants are correlated to one another. Our proposed model type-IIIA superfluid [5,6], rotation, electromagnetic field [7– of anisotropic compact star obeys all the necessary physi- 9] etc. The effect of pressure anisotropic has been studied in cal requirements which have been analyzed with the help details in the Refs. [10,11]. By using uniform matter density, of the graphical representation where n lies in the range of anisotropic model has been investigated in [12]. Chan et al. − 200 ≤ n ≤ 200. We have shown that the model satisfies [13] studied the role of the local pressure anisotropy in detail all the energy conditions as well as the causality condition. and showed that small anisotropies might in principle dras- The model is potentially stable and also satisfy Harrison– tically change the stability of the system. Some anisotropic Zeldovich–Novikov’s stability condition. compact star models are obtained which admit conformal motion. The solutions depend on the conformal factor and matter density ρ [14,15] or a relation between radial and 1 Introduction transverse pressure etc. [14]. In Ref. [15], a new class of interior solutions for anisotropic stars are obtained by choos- A well known fact is that the compact objects such as white ing a particular density distribution function of Lorentzian dwarfs, neutron stars or black hole are formed at the end type as provided by Nazari and Mehdipour [16,17] which point of the gravitational collapse of a star when most of their admits conformal motion in higher dimensional noncommu- nuclear fuel has been consumed. They are supported by either tative spacetime. Some researchers obtained the polytropic nuclear or electromagnetic forces which act as an internal quark star model [21] in order to establish a relation between pressures, but gravity plays an important role in both the theory and observations. Heintzmann and Hillebrandt [19] process of the gravitation collapse and in the final equilibrium described a model of a relativistic, anisotropic neutron star configuration. The factor which is reasonable behind the type model at high densities by means of several simple assump- of the ends up object is the star’s mass. White dwarfs are tions and have shown that for an arbitrary large anisotropy formed from light stars with masses M < 4M and radius is there is no limiting mass for neutron stars, however the maxi- − ∼ 10 2 R Moreover it has shown that the maximum allowed mum mass of a neutron star still lies beyond 3−4M. Sharma mass for white dwarfs is around 1.4M.[1](M, R are et al. [20] considered the theoretical possibility of anisotropy the solar mass and solar radius respectively). in strange stars, with densities greater than that of neutron Ruderman [2] was the pioneer who proposed that the pres- stars but less than that of black holes. Lai and Xu [21]pro- sure inside a highly compact objects shows anisotropic in posed that the polytropic equations of state are stiffer than the nature, i.e., it has two components: one is radial pressure pr conventional realistic models, i.e., the MIT bag model. In the framework of a polytropic model , they show that a very low a e-mail: [email protected] massive quark star can also exist and be still gravitationally 123 138 Page 2 of 13 Eur. Phys. J. C (2019) 79 :138 stable even if the polytropic index n > 3. The properties of 1 Rij − Rgij = Gij, (3) compact stars depends on the assumed description of matter 2 in their interiors. Xu et al. [22] and Azam et al. [23] studied where Rij, R and gij are the Ricci tensors and Ricci scalar the behavior and the physical properties of several compact and metric tensor respectively. Tij is the energy-momentum objects. A general scheme for compact astrophysical objects tensor of the underlying fluid distribution. which are not composed of neutron matter, but where, given Let us assume that the matter involved in the distribution the conditions of very high density in their interiors are pre- is anisotropic in nature, by using the general expression, we, sented by Alcock et al. [24] and Haensel et al. [25]. therefore, get the expression for energy-momentum tensor as The Randall–Sundrum (RS) second brane-world model follows: [26] is stand on the concept that our 4-dimensional space- μ μ μ μ μ μ T = ρv vξ + p χξ χ + p (v vξ − χξ χ − g ), (4) time is a hypersurface embedded into the 5-dimensional ξ r t ξ μ μ bulk. After the work done by Randall and Sundrum [26]on with v vξ = 1 =−χξ χ , χξ is the unit space-like vector brane theory, the study on embedding spacetime attracted the and vμ is the fluid-4 velocity of the rest frame and there- μ researchers more. If a n-dimensional space Vn can be embed- fore v χξ = 0. The above formula gives the components of ded in (n+k)-dimensional space, where k is a minimum num- the energy-momentum tensor of an anisotropic fluid at any ber of extra dimensions, then Vn is said to be of embedding point in terms of the density ρ, the anisotropic and transverse class k of n-dimensional space. Two very well known impor- pressures pr , pt respectively. With the simple form of line μ tant solutions, e.g. Friedmann universe and Schwarzschild’s element, Tξ takes the form: interior solutions are of class I, (so in this case k = 1), on 0 = ρ, 1 =− , 2 = 3 =− , the other hand, Schwarzschild exterior solution is of class II T0 T1 pr T2 T3 pt (5) ( = ) ( = ) k 2 and the Kerr metric [27]isofclassV k 5 .The and Karmarkar condition relates to class one spacetimes. Pandey k = = , and Sharma [39] showed that the Karmarkar condition is Tj 0 if j k only a necessary condition for a spacetime to be of class 1. using (5), for the line element (1), (2) takes the form: A further requirement has to be imposed for sufficiency of −λ −λ the Karmarkar condition. The derivation of the Karmarkar e − 1 e ν πp = + , 8 r 2 (6) Both charged and uncharged star model of embedding class- r r 2 I spacetime extensively study in the Refs. [28–38]. −λ ν ν ν λ ν − λ 8πpt = e + − + , (7) We have organized the paper as follows: in Sect. 2 the 2 4 4 2r basic field equations have been discussed, in Sect. 3 we have 1 − e−λ e−λλ given a short discussion about embedding class-I spacetime 8πρ = + , (8) r 2 r and also obtained a new model. In next section we match our interior space-time with the exterior Schwarzschild line ele- where differentiations with respect to r are denoted by ‘ = = ment and junction condition is also discussed. The physical and we have chosen G c 1. Here G is the gravitational analysis of the model are discussed in Sect. 5. In Sect. 6 we constant and c is the speed of light. The gravitational mass in a sphere of radius ‘r’ is given by, discussed the stability conditions of the present model and some discussions are made in the final section. r m(r) = 4π ρ(ω)ω2dω. (9) 0 In next section we shall solve the Eqs. (6)–(8) to obtain the 2 Basic field equations model of compact star. The interior of a static spherically symmetric spacetime in standard co-ordinate xa = (t, r,θ,φ)is described by the 3 A particular model following line element: So form the above field equations (6)–(8), it is noted that ν( ) λ( ) ds2 = e r dt2 − e r dr2 − r 2(dθ 2 + sin2 θdφ2), (1) we have three field equations for five unknown functions: ν λ e , e ,ρ,pr and pt . So to generate a model of compact where λ and ν are the functions of radial coordinate r only. star, we are free to chose any two of them, though to be a The Einstein field equation is, physically realistic model a number of physical conditions to be satisfied by our present model. Gij = 8πTij. (2) To generate a particular model of compact star, let us 2 Here Gij is the Einstein’s tensor having the following expres- assume that the co-efficient of dr , i.e., grr has the following sions, form: 123 Eur.