Dimensional Charged Gravastars ∗ Farook Rahaman A, ,A.A.Usmanib,Saibalrayc, Saﬁqul Islam A

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Physics Letters B 717 (2012) 1–5

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Physics Letters B

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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

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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 spherical 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.

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. Interior region 4πr where a ‘ ’ denotes differentiation with respect to the radial pa- Seeking interior solution which is free of any mass-singularity =− rameter r.Eq.(12) can equivalently be expressed in the form at the origin, we use the assumption p ρ iteratively. We note that this type of equation of state is available in the literature and r is known as a false vacuum, degenerate vacuum, or ρ-vacuum [29– 4π λ(r) q(r) E(r) = rσ (r)e 2 dr = , (13) 32] and represents a repulsive pressure. Hence by using the result r r given in Eq. (16), we obtain the following interior solutions: 0 F. Rahaman et al. / Physics Letters B 717 (2012) 1–5 3

− + 2B + eγ = e λ = C − 8Ar2 R2n 2 + r2n 4 − Λr2, (21) Integrate (25) immediately to yield (n + 2)(2n + 2) 2B + 2 + h = D − 2Λr2 − r2n 4, (27) 2πσ (2n 4) + + + ρ =−p = 0 R2n 2 − r2n 2 , (22) n 2 + + 2 (2n 2)(n 2) where D is an integration constant. Since, in the interior of the ≈ + 16π 2σ 2 shell, h takes very very small values, one can assume h 0( ), where A = B(2n 4) and B = 0 .HereC is an integration con- 8(2n+2) (n+2)2 where is the thickness of the shell. We further assume that thick- stant. ness of the shell is very very small. This means is very very We assume the surface of the charged distribution, i.e. shell small, 0 < 1. As a result, we demand that D, Λ and B all = is located at r R. Therefore, we have the boundary condition should be 1. In other words, we can take D, Λ and B are all of = p(R) 0. Here, we find the active gravitational mass M(r) in the order of ,i.e.≈ 0( ). following form Employing this value in Eq. (26), we obtain the other metric coefficient as R 2 E γ 1 M(r) = 2πr ρ + dr e = . (28) + 1 8π [Λ + Br2n 2] n+1 0 Also, using TOV equation (20),onecanget 4(2n2 + 8n + 6)π 2σ 2 0 2n+4 = R . (23) n + 2 + (n + 2)2(2n + 2)(2n + 4) 8π p = 8πρ = Λ + Br2n 2 . (29) n It can be noted from Eq. (21) that, C being a non-zero integra- Unlike the interior region, we note that the cosmological con- tion constant, the space–time metric thus obtained is free from any stant Λ has a definite contribution to the pressure and density central singularity. It can also be observed via Eqs. (22) and (23) parameters of the shell in an additive manner. that the physical parameters, viz. density, pressure and mass, are dependent on the charge. Therefore the solutions provide electro- 6. Proper length and energy magnetic mass model, such that for vanishing charge density σ all the physical parameters do not exist [33–44]. However, in this con- We assume the interfaces at r = R and r = R + describing nection one interesting point we note that for the interior region the joining surface between two regions I and III. Recall the join- the above mentioned physical parameters in no way are dependent ing surface is very thin. Now, we calculate the proper thickness on the cosmological constant Λ. between two interfaces, i.e. of the shell as: R+ R+ 4. Exterior region of charged gravastar 1 = eλ dr = √ dr. (30) h(r) The electrovacuum exterior (p = ρ = 0) solution corresponds to R R a static, charged BTZ black hole is written in the following form √ 1 dF = √ 1 Let F be the primitive of . Then, dr . as [27] h(r) h(r) Hence, we get 2 =− − − 2 − 2 2 =[ ]R+ ds M0 Λr Q ln r dt F R . (31) −1 Using Taylor’s theorem, one can expand F (R + ) about R and re- + −M − Λr2 − Q 2 ln r dr2 + r2 dθ 2. (24) 0 taining terms up to the first order of , then, F (R + ) F (R) + F (R) and our would be ≈ dF | = . The parameter M0 is the conserved mass associated with dr r R asymptotic invariance under time displacements. This mass is Therefore, the expression for can be written as given by a flux integral through a large circle at space-like infinity. ≈ . (32) The parameter Q is total charge of the gravastar. − 2 −[ 2B ] 2n+4 D 2ΛR n+2 R 5. Shell The real and positive value of proper length implies D > 2ΛR2 + [ 2B ] 2n+4 n+2 R . However, it can be noted that the thickness between We consider thin shell contains ultra-relativistic fluid of soft two interfaces becomes infinitely large if D takes the value very = 2 +[ 2B ] 2n+4 quanta obeying equation of state p ρ. This assumption is not close to 2ΛR n+2 R . On the other hand the proper length new rather known as a stiff fluid and this type of equation of state will decrease in the absence of the cosmological constant Λ.Ob- which refers to a Zel’dovich Universe have been used by various viously, the estimated size , i.e. proper thickness and√ coordinate authors to study some cosmological [45–47] as well as astrophysi- thickness of the shell are different. Actually, ≈ o( ) as D, Λ cal [48–50] phenomena. and B ≈ o( ). It is very difficult to solve the field equations within the non- We now calculate the energy E˜ within the shell only and we vacuum region II, i.e. within the shell. However, one can obtain get analytic solution within the framework of thin shell limit, 0 < + − R e λ ≡ h 1. The advantage of using this thin shell limit is that E2 E˜ = 2π ρ + rdr in this limit we can set h to be zero to the leading order. Then the 8π = field equations (9)–(11),withp ρ, may be recast in the forms R + 1 (2n 4)Λ 2 2 =− + 2 = (R + ) − R h 4r Λ E , (25) 4 4n γ (4n + 4)B + + h = 2E2. (26) + (R + )2n 4 − R2n 4 . 4 2n(2n + 4) 4 F. Rahaman et al. / Physics Letters B 717 (2012) 1–5

Since the thickness of the shell is very small, i.e. 1, we ex- So, the discontinuity in the second fundamental forms is given as pand it binomially about R and taking first order of ,weget + − κij = K − K . (39) + + ij ij ˜ (2n 4)ΛR (4n 4)B 2n+3 E ≈ + R . (33) Now, from Lanczos equation in (2 + 1)-dimensional space–time, 4 2n 2n the field equations are derived [52]: Obviously here n = 0. As before, we also notice that the energy E˜ 1 φ is of the order of 2,i.e.E˜ ≈ o( 2). σ =− κ , (40) 8π φ 1 7. Entropy v =− κτ , (41) 8π τ Adopting concept of Mazur and Mottola [1,2],wetrytocalcu- where σ and v are line energy density and line tension. Employing late the entropy of the fluid within the shell relevant information into Eqs. (40) and (41), and also by setting r = R,weobtain R+ 1 λ 2 2 S = 2π s(r)r e dr. (34) σ =− −ΛR − M0 − Q ln R 8π R R 2B(2n + 3) Here s(r), the entropy density for the local temperature T (r),is + C − 8ΛR2n+4 − R2n+4 , (42) + + given by (2n 4)(2n 2) 2 −ΛR − Q α2k2 T (r) k p 1 2R = B = B v =− s(r) α , (35) 8π − 2 − − 2 4πh¯ 2 h¯ 2π ΛR M0 Q ln R − 2n+3 − B(2n+3) 2n+3 2 ( 8ΛR (2n+2) R ) where α is a dimensionless constant. + . (43) Thus the entropy of the fluid within the shell, via Eq. (29),be- 2n+4 2B(2n+3) 2n+4 C − 8ΛR − + + R comes (2n 4)(2n 2) Similar to the (3 + 1)-dimensional case the energy density is neg- R+ + 2n 4 [Λr2 + Br2n+4] dr kB 1 2n ative in the junction shell. It is also noted that the line tension is S = 2π α . (36) ¯ negative which implies that there is a line pressure as opposed to h 4π D − 2 r2 − 2B r2n+4 R Λ n+2 a line tension. The thin shell, i.e. region II of our configuration con- tainstwotypesoffluidnamely,theultra-relativisticfluidobeying Since, thickness of the shell is negligibly small compared to its po- p = ρ and matter component with above stress tensor components sition R from the centre of the gravastar (i.e. R), in the similar (42) and (43) that are arisen due to the discontinuity of second way, as we have done above, by expanding the primitive of the fundamental form of the junction interface. above integrand about R and retaining terms up to the first order These two non-interacting components of the stress energy ten- of ,wehavetheentropyas sors are characterizing features of our non-vacuum region II.

+ + 2n 4 [ΛR2 + BR2n 4] 9. Concluding remarks kB 2n S ≈ α . (37) h¯ 2 − 2 − 2B 2n+4 D 2ΛR n+2 R In this Letter, we have presented a new model of charged gravastar in connection to the electrovacuum exterior (2 + 1) anti- The expression for the entropy shows that the cosmological de Sitter space–time. One of the most interesting features of this constant Λ contributes to it a constant part. The entropy of the model is that it is free from any singularity and hence represents shell is of the order of ,i.e.S ≈ 0( ). a competent candidate in the class of gravastar as an alternative to black holes [1,2]. 8. Junction condition The solutions obtained here represent electromagnetic mass model [35].Historically,Lorentz[33] proposed his model for ex- To match the interior region to the exterior electrovacuum so- tended electron and conjectured that “there is no other, no ‘true’ or lution at a junction interface S,situatedatr = R, one needs to ‘material’ mass”, and thus provides only ‘electromagnetic masses of use extrinsic curvature of S. The surface stress energy and sur- the electron’ whereas Wheeler [57] and Wilczek [58] pointed out face tension of the junction surface S could be determined from that electron has a “mass without mass”. Later on several works the discontinuity of the extrinsic curvature of S at r = R.Herethe have been carried out by different investigators [34,36–44] under junction surface is a one-dimensional ring of matter. Let η denote the framework of general relativity. the Riemann normal coordinate at the junction. We assume η be The cosmological constant Λ as proposed by Einstein [59] for positive in the manifold in region III described by exterior elec- stability of his cosmological model has been adopted here as a trovacuum BTZ space–time and η be negative in the manifold in purely scalar quantity which has a definite contribution to the region I described by our interior space–time. In terms of math- physical parameters with a constant additive manner. However, ematical notations, we have xμ = (τ ,φ,η) and the normal vector due to accelerating phase of the present Universe [60,61],this μ components ξ = (0, 0, 1) with the metric gηη = 1, gηi = 0and erstwhile Λ is now-a-days considered as a dynamical parameter 2 gij = diag(−1, r ). The second fundamental forms associated with varying with time in general. It is therefore a matter of curious is- the two sides of the shell [51–56] are given by sue whether varying Λ has any contribution in formation of the gravastars. 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