View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Physics Letters B 717 (2012) 1–5 Contents lists available at SciVerse ScienceDirect Physics Letters B www.elsevier.com/locate/physletb The (2 + 1)-dimensional charged gravastars ∗ Farook Rahaman a, ,A.A.Usmanib,SaibalRayc, Safiqul Islam a a Department of Mathematics, Jadavpur University, Kolkata 700 032, West Bengal, India b Department of Physics, Aligarh Muslim University, Aligarh 202 002, Uttar Pradesh, India c Department of Physics, Government College of Engineering and Ceramic Technology, Kolkata 700 010, West Bengal, India article info abstract Article history: This is a continuation and generalization of our earlier work on gravastar in (2 + 1) anti-de Sitter Received 30 May 2012 space–time to (2 + 1)-dimensional solution of charged gravastar. Morphologically this gravastar contains Received in revised form 28 July 2012 three regions, namely: (i) charged interior, (ii) charged shell and (iii) electrovacuum exterior. We have Accepted 6 September 2012 studied different characteristics in terms of Length and Energy, Entropy, and Junction conditions of the Available online 10 September 2012 spherical charged distribution. It is shown that the present model of charged gravastar is non-singular Editor: M. Trodden and represents itself an alternative of black hole. © Keywords: 2012 Elsevier B.V. Open access under CC BY license. Gravitation Equation of state Charged gravastar 1. Introduction space–time was (3 + 1)-dimensional. The outer region of the Ra- haman et al. [17] model of gravastar corresponds to the exte- Recently the study of gravastars, the gravitational vacuum star, rior (2 + 1) anti-de Sitter space–time of BTZ-type black holes as has become a subject of considerable interest as it was proposed presented by Bañados, Teitelboim and Zanelli [18]. Therefore, the as an alternative to black holes. Mazur and Mottola [1,2] first pro- above two works demand that one should investigate a (2 + 1)- posed a new type of solution for the endpoint of a gravitational dimensional solution for charged gravastar. This is the motivation collapse in the form of cold, dark and compact objects. Therefore, of our present investigation. physically this was an extension of the concept of Bose–Einstein In favour of inclusion of charge in stellar distribution it has condensate to gravitational systems. The Mazur–Mottola model [1, been argued in the work of Usmani et al. [16] that compact stars 2] contains an isotropic de Sitter vacuum in the interior, while tend to assemble a net charge on the surface [19–23]. This facil- the exterior is defined by a Schwarzschild geometry, separated by itates stability of a fluid sphere by avoiding gravitational collapse a thin shell of stiff matter implying that the configuration of a and hence singularity [19,20,24–26]. In this connection we would gravastar has three different regions with different equations of like to mention the interesting charged model of Horvat [13] which state (EOS) [3–15], designated by: (I) Interior: 0 r < r1, p =−ρ; represents a gravastar where the analysis has carried out within Is- (II) Shell: r1 < r < r2, p =+ρ; and (III) Exterior: r2 < r, p = ρ = 0. rael’s thin shell formalism and the continuous profile approach. It is argued that the presence of matter on the thin shell of thick- In the present investigation we have followed the mechanism ness r2 − r1 = δ is essential to achieve the required stability of of Mazur and Mottola [1,2] in the framework of Einstein–Maxwell systems under expansion by exerting an inward force to balance formalism. To solve the specified equations, related to the regions the repulsion from within. designated as Interior, Shell and Exterior, we have considered ap- Usmani et al. [16] proposed a new model of a gravastar ad- propriate equations of state (EOS), viz. ρ =−p (dark energy), mitting conformal motion by assuming a charged interior. Their ρ = p (stiff fluid) and ρ = p = 0 (dust) respectively. Under this re- exterior was defined by a Reissner–Nordström line element instead gional classification of the spherical configurations we have solved of Schwarzschild’s one. Later on Rahaman et al. [17] designed a the Einstein–Maxwell field equations in the specific cases. There- neutral spherically symmetric model of gravastar in (2 + 1) anti- after we have characterized the interfaces and shell in terms of de Sitter space–time contrary to the former work where, as usual, Length, Energy and Entropy. The Junction conditions of the spher- ical charged distribution are imposed on the different regions by using Lanczos equations in (2 + 1)-dimensional space–time. The model thus obtained represents an alternative to black hole in * Corresponding author. E-mail addresses: [email protected] (F. Rahaman), [email protected] the form of charged gravastar as it is free from any singular- (A.A. Usmani), [email protected] (S. Ray), sofi[email protected] (S. Islam). ity. 0370-2693 ©2012ElsevierB.V. Open access under CC BY license. http://dx.doi.org/10.1016/j.physletb.2012.09.010 2 F. Rahaman et al. / Physics Letters B 717 (2012) 1–5 2. Einstein–Maxwell equations where q(r) is total charge of the sphere under consideration. For a charged fluid distribution, the generalized Tolman–Oppenheimer– We first consider the line element for the interior space–time Volkov (TOV) equation may be written as of a static spherically symmetric charged distribution of matter in 1 1 (2 + 1) dimensions in the form [27,28] (ρ + p)γ + p = r2 E2 , (14) 2 8πr2 2 γ (r) 2 λ(r) 2 2 2 ds =−e dt + e dr + r dθ . (1) which is the conservation equation in (2 + 1) dimensions. We note that the term inside the integral sign in Eq. (13) is The Hilbert action coupled to electromagnetism is given by λ(r) σ (r)e 2 , which is equivalent to the volume charge density. We √ − 3 R 2Λ 1 c will consider the volume charge density in polynomial function I = dx −g − F Fbc + Lm , (2) 16π 4 a of r. Hence we use the condition where L is the Lagrangian for matter. The variation with respect λ(r) n m σ (r)e 2 = σ0r , (15) to the metric gives the following self-consistent Einstein–Maxwell equations with cosmological constant Λ for a charged perfect fluid where n is arbitrary constant as polynomial index and the con- distribution stant σ0 is referred to the central charge density. By using the latter result of Eq. (15),oneobtainsfromEq.(13) 1 PF EM as Rab − Rgab + Λgab =−8π T + T . (3) 2 ab ab 4πσ0 n+1 The explicit forms of the energy momentum tensor (EMT) compo- E(r) = r , (16) n + 2 nents for the matter source (we assumed that the matter distribu- 4πσ0 + tion at the interior of the star is perfect fluid type) and electro- q(r) = rn 2. (17) magnetic fields are given by n + 2 Now, we write some consequences of the filed equations and TOV T PF = (ρ + p)u u + pg , (4) ab i k ik equation. Eq. (9) implies 1 1 EM c cd −λ(r) 2 T =− F Fbc − gab Fcd F , (5) e = M(r) − Λr , (18) ab 4π a 4 where where ρ, p, ui and Fab are, respectively, matter-energy density, fluid pressure and velocity three vector of a fluid element and r r electromagnetic field. Here, the electromagnetic field is related to M(r) = C − 16π rρ(r) dr − 2 rE2(r) dr (19) current three vector 0 0 J c = σ (r)uc, (6) is the active gravitational mass of the spherical distribution. Since, eλ(r) > 0 within the charged sphere of radius R as well as regular as at the origin, we demand that C > 0. Using Eqs. (10), (13) and the ab =− a above result (18), we finally obtain the TOV equation as F;b 4π J , (7) 2 (p + ρ)(8π p − E − Λ)r 1 where, σ (r) is the proper charge density of the distribution. In p =− + q2 . (20) = t M(r) − Λr2 8 r2 our consideration, the three velocity is assumed as ua δa and π consequently, the electromagnetic field tensor can be given as Following Mazur and Mottola [1,2] we consider a new (2 + 1)- dimensional charged perfect fluid configuration which has three = t r − r t Fab E(r) δaδb δaδb , (8) different regions with different equations of state: where E(r) is the electric field. =− The Einstein–Maxwell equations with a cosmological constant IInterior:0 r < R, ρ p; + = (Λ<0), for the space–time described by the metric (1) together II Shell: R r < R , ρ p; = = with the energy–momentum tensor given in Eqs. (4) and (5),yield III Exterior: r2 < r, ρ p 0. (rendering G = c = 1) Accordingly our configuration is supported by an interior region −λ λ e with equation of state p =−ρ. The shell of our configuration be- = 8πρ + E2 + Λ, (9) 2r longs to the interfaces, r = R and r = R + , where is the thick- − γ e λ ness of the shell. This thin shell, where the metric coefficients are = 8π p − E2 − Λ, (10) continuous, contains ultra-relativistic fluid of soft quanta obeying 2r equation of state ρ = p. The outer region of this gravastar corre- −λ e 1 1 γ 2 + γ − γ λ = 8π p + E2 − Λ, (11) sponds to the electrovacuum exterior solution popularly known as 2 2 2 the charged BTZ black hole space–time. − λ e 2 σ (r) = (rE) , (12) 3.