Chinese Physics C Vol. 42, No. 11 (2018) 115101
Stable gravastars: Guilfoyle’s electrically charged solutions
Ayan Banerjee1;1) J. R. Villanueva2;2) Phongpichit Channuie3;3) Kimet Jusufi4;4) 1 Department of Mathematics, Jadavpur University, Kolkata-700032, India 2 Instituto de F´ısica y Astronom´ıa, Facultad de Ciencias, Universidad de Valpara´ıso, Gran Breta˜na 1111, Valpara´ıso, Chile 3 School of Science, Walailak University, Nakhon Si Thammarat, 80160 Thailand 4 Physics Department, State University of Tetovo, Ilinden Street nn, 1200, Macedonia
Abstract: Compelling alternatives to black holes, namely, gravitational vacuum stars (gravastars), which are mul- tilayered compact objects, have been proposed to avoid a number of theoretical problems associated with event horizons and singularities. In this work, we construct a spherically symmetric thin-shell charged gravastar model where the vacuum phase transition between the de Sitter interior and the external Reissner–Nordstr¨om spacetime (RN) are matched at a junction surface, by using the cut-and-paste procedure. Gravastar solutions are found among the Guilfoyle exact solutions where the gravitational potential W 2 and the electric potential field φ obey a particular 2 relation in a simple form a(b−ǫφ) +b1, where a, b and b1 are arbitrary constants. The simplest ansatz of Guilfoyle’s Q2 solution is implemented by the following assumption: that the total energy density 8πρm + r4 is constant, where Q(r) is the electric charge up to a certain radius r. We show that, for certain ranges of the parameters, we can avoid the horizon formation, which allows us to study the linearized spherically symmetric radial perturbations around static equilibrium solutions. To lend our solution theoretical support, we also analyze the physical and geometrical properties of gravastar configurations.
Keywords: gravastar, Guilfoyle exact solutions, stability analysis PACS: 04.20.Jb DOI: 10.1088/1674-1137/42/11/115101
1 Introduction ations are expected to play a non–trivial role at or near the event horizon. Based on the gravastar framework, The final state of a star’s gravitational collapse can the solutions of Mazur and Mottola’s gravastar scenario lead to the formation of different objects such as neu- describe five layers of the gravastar, including two thin tron stars, white dwarfs, black holes and naked singu- shells. The de Sitter geometry in the interior, with an larities [1–5]. This end-state of collapse is a widely ac- equation of state (EOS) p = ρ, matches to an exterior − cepted field of research in the scientific community from Schwarzschild vacuum geometry. Between the interior many perspectives, both theoretical and observational. and exterior geometry, there is a finite–thickness shell However, classical general relativity suffers from some comprised of stiff fluid matter with EOS p = +ρ. Due severe theoretical problems. One problem is related to to their extreme compactness, however, it would be very the paradoxical features of black holes and naked sin- difficult to distinguish gravastars from black holes. As gularities. In order to resolve these issues, the idea of an extension of the Mazur–Mottola model and for physi- the existence of compact objects without event horizons cal reasons, Visser and Wiltshire [9] reduced the number has recently been proposed, the so-called gravastar: an of thin shells from 5 layers to 3 layers with a continuous alternative to black holes [6–8]. Moreover, some inter- layer of finite thickness, where the phase transition layer esting articles have been published. Within these mod- was replaced by a single spherical δ-shell. In the model, els, a massive star in its final stages could end its life by the definition of the gravastar, a de-Sitter spacetime as a gravastar that compresses matter within the grav- was matched to a Schwarzschild exterior solution at a itational radius r = 2 GM/c2, i.e., very close to the junction surface with surface stresses σ and . More- s P Schwarzschild radius, but with no singularity or event over, the authors provided full dynamic stability against horizon. In this situation, the quantum vacuum fluctu- spherically symmetric perturbations by using the Israel
Received 7 May 2018, Revised 5 August 2018, Published online 26 September 2018 1) E-mail: ayan [email protected] 2) E-mail: [email protected] 3) E-mail: [email protected] 4) E-mail: kimet.jusufi@unite.edu.mk ©2018 Chinese Physical Society and the Institute of High Energy Physics of the Chinese Academy of Sciences and the Institute of Modern Physics of the Chinese Academy of Sciences and IOP Publishing Ltd
115101-1 Chinese Physics C Vol. 42, No. 11 (2018) 115101 thin shell formalism [10] in terms of an effective energy tral pressure goes to infinity Guilfoyle’s stars also obey equation. This simplified form of gravastar structure has the Buchdahl-Andr´easson bound. Moreover, regarding also motivated a number of physicists to consider differ- the Weyl and Weyl–Guilfoyle relations, general relativis- ent types of vacuum geometries in the interior and ex- tic charged fluids with non–zero pressure turned out to terior. Among these models, Bili´cand his collaborators be an important cornerstone for addressing and discov- have shown how a gravastar structure can form a Born– ering new solutions. Thus, it is interesting to embark Infeld scalar field [11] and a non-linear electrodynamic on a study of the Guilfoyle model with the presence of gravastar [12]. In Ref. [13], gravastar solutions have been electrically charged matter. We expect that gravastar studied by replacing the δ–shell with a continuous stress- solutions should be found within a certain range of the energy tensor from the asymptotically de Sitter interior parameters of the model. to the exterior Schwarzschild solution. Moreover, an The outline of the paper is as follows. In Section 2, electrically charged gravastar has been studied by solving we present Guilfoyle’s exact solutions with the ansatz 2 2 the Einstein–Maxwell field equations in the asymptoti- W = a(b ǫφ) + b1. In Section 3, the basic equa- cally de Sitter interior [14], whereas a charged gravastar tions for spherically− symmetric spacetime with electri- admitting conformality was constructed in Ref. [15]. In cally charged perfect fluid matter distribution are writ- the same vein, Chan et al. [16] have studied radiat- ten, satisfying a particular Weyl–Guilfoyle relation, and ing gravastars by considering Vaidya exterior spacetime. we discuss the interior and exterior solutions with appro- However, it was shown that the interior de Sitter space- priate boundary conditions in Section 4. In Section 5, time may also be replaced by considering a solution gov- we briefly review some gravastar models. In Section 6, erned by the dark energy equation of state, ω=p/ρ where we present the basic setup for matching the two distinct ω< 1/3 [17]. Furthermore, much effort has been made spacetimes and obtaining models of the thin shell gravas- to investigate− the properties of gravastars in the context tar. We investigate the constraints on parameters at the of alternative scenarios. These were considered in Refs. junction interference with a time-like thin shell in Section [18–22]. 7. In Section 8, we obtain the static gravastar solution, An interior regular charged perfect fluid solution can and briefly outline the linearized stability analysis with not only contribute to a better understanding of the the determined stability regions of the transition layer in structure of spacetime, but physically it also offers many Section 9. We also investigate the stability of the gravas- new and interesting solutions. A static solution around tar by using the surface mass of the thin shell, and discuss a spherically non-rotating charged body was stimulated in detail some interesting observations. Our conclusions by the work of Reissner and Nordstr¨om [23]. Almost are given in Section 10. Throughout this work, if not at the same time, Hermann Weyl [24, 25] studied vac- explicitly stated otherwise, we use the units of G=c=1. uum general relativity and electromagnetism, which to- gether established a relation between the metric com- 2 Structural equations of Weyl– ponent gtt and the electric potential φ. A further dis- Guilfoyle charged fluid cussion about this relation appeared in 1947 when Ma- jumdar [26] generalized this result to systems without This section is devoted to presenting the basis equa- spatial symmetry. Regarding this point of view, fur- tions governing the dynamical behavior of a cold charged ther development on Weyl’s work was examined much gravitating distribution of a relativistic fluid. The start- later by Guilfoyle [27]. He considered charged fluid dis- ing point is to revise the Einstein–Maxwell field equa- tributions in which the interior of these solutions is char- tions in a four-dimensional spacetime given by acterized by the relation between the gravitational and 1 the electric potential, which are functionally related to G =R g R=8π(T +E ), (1) µν µν −2 µν µν µν each other via W = W (φ). Here W is parametrized by W 2 =a(b ǫφ)2+b , where a, b and b are arbitrary con- and, − 1 1 stants, and ǫ = . Specifically, this relation generalizes µν µ ν F =4πJ , (2) the common Weyl± relation, when a = 1, and the exten- ∇ F =0 (3) sion of this concept was discussed by Lemos and Zanchin ∇[α µν] [28]. Here they obtained a relationship between various where Rµν is the Ricci tensor, R is the Ricci scalar, and
fields and matter quantities. The authors also studied Tµν is the energy–momentum tensor, for which we as- quasiblack holes such as frozen stars [29], and found reg- sume it is a perfect fluid: ular black holes for the Guilfoyle exact solutions [30] with T =(ρ +p)U U +pg , (4) the discussion of their physical relevances. In addition, µν m µ ν νµ
Lemos and Zanchin [31] applied the method to study rel- where ρm is the energy density, p is the isotropic fluid ativistic charged spheres, and showed that when the cen- pressure, and Uµ is the 4–velocity of the relativistic
115101-2 Chinese Physics C Vol. 42, No. 11 (2018) 115101
matter fluid. Also, Eµν is the energy tensor of the elec- where the structural functions N and B depend on the tromagnetic field, given by radial coordinate r only, so the gauge field and 4–velocity 1 1 are then given by E = F λ F g F F λσ , (5) µν 4π µ νλ−4 µν λσ A = φδ0 , and U = N(r)δ0 . (15) µ − µ µ − µ with Fµν representing the electromagnetic field-strength We now turn our attention to the stellar mass equation tensor, given by of the spherically symmetric solutions due to the total contribution of the matter energy density and electric Fµν =∂µAν ∂ν Aµ, (6) − energy density. Thus, the mass inside a sphere of radius where Aµ = φ,A is the electromagnetic 4–potential. Fi- r can be obtained from the following relation: r 2 2 nally, with ρe denoting the associated density of electric Q Q M(r)=4π ρ (r)+ r2 dr+ , (16) charges, the 4–current density can be expressed as m 8πr4 2r 0 Jµ =ρeUµ, (7) and therefore, the total charge enclosed in the same re- whose temporal component J 0 is equal to the charge den- gion becomes sity ρ , and whose spatial components J i are just the r e Q(r)=4π ρ (r) B(r)r2dr. (17) usual 3–vector current components. A static spherically e 0 symmetric spacetime is described by the line element Our main goal here is to study a type of system for which 2 µ ν ds =gµν dx dx , (8) the lapse function N(r) connects the gravitational and electric potentials via the Majumdar–Papapetrou rela- which can be rewritten in the form 2 2 tion N(r)=W (φ(r))=a[b ǫφ(r)] (b1 =0 in Eq. (11)), 2 2 2 i j − ds = W dt +hij dx dx , (9) in which case one gets − where W is the gravitational potential, so the 4–potential N(r) Aµ and the 4–velocity Uµ are defined via ǫφ(r)=b . (18) − a 0 0 Aµ = φδ , and Uµ = Wδ . (10) Due to the additive nature of the fields, the constant − µ − µ Notice that we are considering only pure electric fields b can be absorbed into the potential, so without loss of A =( φ,0,0,0), and also, h are functions of the spatial generality we choose b=0 and the lapse function becomes µ − ij coordinates xi only with i=1,2,3. In particular, we are N(r)=a[ǫφ(r)]2 =aφ(r)2. (19) interested in a class of the Guilfoyle solutions [27, 29] where the gravitational potential W and the electric po- For static spherically symmetric configurations, as con- tential φ are related by means of the equation sidered in the present research, we have written only the components of tt and rr of the Einstein equation (1), W (φ)= a(b ǫφ)2+b , (11) giving the following relations [28, 29]: − 1 where a, b and b are arbitrary constant and ǫ= 1. Here 1 d 1 ln[N(r)B(r)]=8π(ρm(r)+p(r)), (20) the parameter a is the Guilfoyle parameter. As± stated in rB(r) dr 2 Ref. [28], from the set of quantities ρm,p,ρe,ρem we d r Q { } =8π ρm(r)+ , (21) obtain the following equation of state: dr B(r) 8πr4 a(b ǫφ)ρe ǫW (φ)[ρm+(1 a)ρem] whereas the first integral of the only nonzero component p= − − − , (12) 3ǫW (φ) of Maxwell’s equations (2) gives
2 ′ where ρem is the electromagnetic energy density defined r φ (r) Q(r)= , (22) by means of the Weyl–Guilfoyle relation: N(r)B(r) 2 1 ( iφ) where an integration constant is set to zero and a prime ρem = ∇ . (13) 8π W 2(φ) denotes a derivative with respect to the radial coordinate r. Therefore, by making use of Eq. (19) in Eq. (22), we 3 Spherical equations: general analysis obtain a differential form for the electric charge, ǫ r2N ′(r) The geometry of the static spherically symmetric so- Q(r)= . (23) lution for a charged fluid distribution found by Guilfoyle −2√a B(r)N(r) [27] can be written in the usual Schwarzschild coordi- Therefore, one can easily obtain the electric charge inside nates, (t,r,θ,ϕ) as a spherical surface of radius r, if the metric functions are ds2 = N(r)dt2+B(r)dr2+r2 dθ2+sin2 θdϕ2 , (14) known. Finally, the conservation law T µν =0 together − ∇µ 115101-3 Chinese Physics C Vol. 42, No. 11 (2018) 115101
− with Maxwell’s equations leads to the hydrostatic equi- r2 a r2 m q2 1 c =c 1 0 1 0 . (33) librium equation that determines the global structure of 1 0 2 2 2 −R −(2 a) R r0 −r0 an electrically charged star by requiring the conservation − of mass-energy [32, 33] for the system: Thus, from Eq. (19), the electric potential is given by a ′ ′ (a−2) ′ N (r) φ (r)ρe(r) ǫ (2 a) 2p (r)+ [ρm(r)+p(r)] 2 =0, (24) φ(r)= − F (r) , (34) N(r) − N(r) √a a which is the only non-identically zero component of the whereas the fluid quantities are given by conservation equations. The first two terms on the l.h.s. 3 ac2 r2 8πρ (r)= 1 0 , (35) come from the gravitational force with an isotropic pres- m R2 −3R2 (2 a)2 F 2(r) sure and density, while the second term is due to the − 1 ac2 r2 2a F (r)+c Coulomb force that depends on the matter by the met- 8πp(r)= 1+ 0 + 1 , ric coefficient. −R2 R2 (2 a)2 F 2(r) −(2 a) F (r) − − (36) ǫ√a r2 3F (r)(F (r)+c ) 4 Guilfoyle’s solutions for the interior 8πρ (r)= c2+ 1 . (37) e R4(2 a) F 2(r) 0 r2 region − Finally, inserting the above relations into Eqs. (16) and In this section we describe a static spherically symet- (17), the mass and electric charge functions M(r) and ric distribution of electrically charged matter in the range Q(r) are found to be 0 6 r 6 r0. To do this, we adopt the simplest ansatz of r3 ac2 r2 Guilfoyle’s solution [27] given by M(r)= 1+ 0 , (38) 2R2 R2(2 a)2 F 2(r) Q2 3 − 8πρm+ 4 = 2 , (25) and r R ǫ√ac r3 Q(r)= 0 , (39) where r 6 r0 and R is a constant which characterizes R2 (2 a) F (r) the length associated with the inverse of the total energy − density. It can also be related to the parameters of the respectively. exterior solution, namely the total charge q and the total 5 Review of gravastar models mass m evaluated at the junction boundary r=r0, 2 1 2 q This section presents a quick review of the main fea- 2 = 3 m , (26) R r −2r0 tures of two gravastar models, which have some interest- 0 where ing properties for our subsequent study. r0 Q2 q2 5.1 Mazur–Mottola model m=M(r )=4π ρ (r)+ r2 dr+ . (27) 0 m 8πr4 2r 0 0 The first model presented here is the so–called Mazur q=Q(r0). (28) & Mottola gravastar model [6]. According to this model, Using this additional assumption, Guilfoyle [27] the interior of the gravastar is a de Sitter spacetime sur- found the exact solutions of the Einstein–Maxwell equa- rounded by a layer of ultra–stiff matter, while the exte- tions, which can be summarized as follows. The struc- rior is then suitably matched by a Schwarzschild space- tural functions become time i.e., there are five different regions (including two thin shells), each with its own features: r2 B(r)=1 2 , (29) −R 1) Inside the gravastar 06r 2 2 2 −1/a R m q r 5) An exterior Schwarzschild vacuum r > r2 with c = 1 0 , (32) 0 r2 r −r2 −R2 p=ρ=0. 0 0 0 115101-4 Chinese Physics C Vol. 42, No. 11 (2018) 115101 These features are the most important for a gravastar and B(r) are given by model having negative central pressure, positive density r2 and the absence of event and cosmological horizons. No- 1 when r 115101-5 Chinese Physics C Vol. 42, No. 11 (2018) 115101 ij exterior spacetimes given in Eq. (40), so they become still satisfies an energy conservation law, iS =0, by virtue of i (κij δ κ)=0 at the junction interface.∇ ∇ − ij 2 Now, using two equations, Eqs. (51) and (52), it is µ a˙ a 2 n− = , 1 +˙a ,0,0, (45) easy to check the energy conservation equation is ful- a2 −R2 1 filled: −R2 d d (σa2)+ (a2)=0, (53) dτ P dτ a˙ 2M q2 nµ = , 1 + +˙a2,0,0 , (46) from which immediately follows + 2M q2 − a a2 1 + a˙ − a a2 σ˙ = 2(σ+ ) . (54) − P a where the ( ) superscripts correspond to the exterior and interior± spacetimes, respectively. Integrating out the above equation, it yields The discontinuity of the extrinsic curvature κ can 2 ij σ′ = (σ+ ), (55) be written in a simple form due to spherical symmetry −a P i τ θ θ θ ϕ as κj = diag(κτ ,κθ,κθ), with κθ = κϕ. By employing the where primes and dots denote differentiation with re- expressions through Eq. (42), we find the non–vanishing spect to a and τ, respectively. The first term on the left components of surface stress–energy tensor that can be side of Eq. (53) represents the internal energy change of written in terms of Si = diag( σ, , ), where σ is the j − P P the shell, while the work done by internal forces of the surface energy density and P is the surface pressure. shell is given in the second term. Now, using Eq. (40) and Eq. (43), the non–trivial Now, by taking into account Eqs. (51-52) and substi- components of the extrinsic curvature are given by tuting into Eq. (55), we obtain the following expression, 2 2 − 1 a θ 2 3M 2q 2 K θ = 1 2 +˙a , (47) 1 + +˙a aa¨ 2 a −R ′ 1 2 1+˙a aa¨ σ = − a a − − , (56) 4πa2 2M q2 − a2 a 1 + +˙a2 1 +˙a2 a¨ 2 2 2 − a a −R Kτ − = −R , (48) τ a2 which plays an important role in determining the stabil- 1 +˙a2 ity regions when the static solution a is being consid- −R2 0 ered. 2 According to Refs. [35, 36], the total surface mass of θ+ 1 2M q 2 2 the thin shell is given by ms =4πa σ. For this case the K θ = 1 + 2 +˙a , (49) a − a a total mass of the system M evaluated at a static solution 2 M q a0 is given by (by rearranging Eq. (51)) a¨+ 2 3 Kτ + = a −a , (50) 3 2 2 τ 2 a0 q a0 ms(a0) M = + +ms 1 . (57) 2M q 2 2 2 1 + +˙a 2R 2a0 −R − 2a0 2 − a a where dots denote the derivatives with respect to τ. Note that M(a) is the total active gravitational mass Therefore, using Eqs. (47, 48, 49, 50) in the Lanc- and q(a)2/2a is the mass equivalent to the electromag- zos equations (42), we find that the energy density netic field. θ σ κθ/4π and the pressure at the junction surface ≡(κ −τ +κθ)/8π become 7 Junction conditions P≡ τ θ 2 2 1 2M q 2 a 2 Keep in mind that a gravastar model does not possess σ= 1 + 2 +˙a 1 2 +˙a , (51) −4πa − a a − −R an event horizon. For instance, if the thin–shell transi- tion layer Σ is located at r=a(τ), then to avoid horizon and formation we demand M 2a2 2 1 +˙a2+aa¨ 1 +˙a2+aa¨ r 2M q 1 2 0< <1 and 0< <1. (58) = − a − R , (52) R r −r2 P 8πa 2M q2 − a2 1 + +˙a2 1 +˙a2 For the present analysis, one can obtain the solution of 2 2 − a a −R the non–rotating thin shell gravastar when the space- respectively. Notice that the surface density σ has the times given by the metrics (40) are matched at a. opposite sign to the surface pressure , and also that all The restriction of charge–to–mass ratio q /M < 1 for the energy–momentum that plungesP into the thin shell the Reissner-Nordstr¨om spacetime corresponds| | to two 115101-6 Chinese Physics C Vol. 42, No. 11 (2018) 115101 horizons, namely the Cauchy and event horizons r± = and M 1 1 q2/M 2 . When q /M 1, these are glued y2 ± − | | → 1 2 into a single horizon. For the case when q /M >1, it is a 1 x 1 x − x2 | | (x)= − . (64) naked singularity. Moreover, we consider the case when P 8πR y √1 2x+w2 x2 − y2 − 1 q /M 6 1 in order to avoid the horizon from geometry, −x2 | | where a>r =M 1+ 1 q2/M 2 . We develop the rest + − of the section by assuming q /M 6 1, where the junction surface r=a is situated outside| | the event horizon. 8 Static gravastars Now, we resolve the static case, which is given by a˙ =¨a=0. In this case, Eqs. (51) and (52) reduce to 2 2 1 2M q a0 σ(a0)= 1 + 2 1 2 , (59) −4πa0 − a0 a0 − −R M 2a2 1 1 0 1 − a0 − R2 (a0)= . (60) 2 P 8πa0 2 − a 2M q 1 0 Fig. 1. Unification diagram for surface energy den- 1 + 2 2 − a a −R sity σ(a0). We have considered the dimensionless 0 0 parameters x=M/a0 and y=M/R. The qualita- It is worth noting that the gravastar solution is also con- tive values used for drawing the graphs are w=0.5 sidered a self-gravitating object, which can avoid the for- and R=3. Note that surface energy density and mation of the black hole horizon. Hence, a gravastar surface pressure are both positive in the range. model can be considered as one type of compact object See the Appendix for more details. where the surface redshift is an important source of in- formation. We expect that the redshifts of gravastars would be higher than any ordinary objects. The surface gravitational redshift is defined by z = ∆λ/λe = λ0/λe, where ∆ is the fractional change between the observed wavelength λ0 and the emitted wavelength λe. Thus, in our notation, the redshift factor za0 is −1/2 z = 1+ g (a ) , (61) a0 − tt 0 Now, the behavior of the surface redshift is not larger than z=2 [37] for a static perfect fluid sphere. This value may increase up to 3.84, when we consider anisotropic fluid spheres [38]. In order to specify an equilibrium solution for the thin–shell gravastar, we introduce the dimensionless con- figuration variables defined by Ref. [39], M M q x , y , and w . (62) Fig. 2. Surface pressure, with w = 0.5 and R = 3 ≡ a0 ≡ R ≡ M used as the values of the parameters for graphical Therefore, assuming the condition of q 115101-7 Chinese Physics C Vol. 42, No. 11 (2018) 115101 the linearized stability of the solutions. In order to test where whether the equilibrium solution is stable or not, we re- ∆ = 256a6π4σ5+a2ζ2σ, arrange Eq. (51) in the following suggestive form, 1 − 5 4 5 2 ′ 2 ∆2 = 1024a π σ +4a σζζ 4aζ σ, 1 2 − − a˙ +V (a)=0, (65) 6 4 4 2 2 2 ∆3 = 256a π σ 3a ζ . which is known as the thin–shell equation of motion, with − − In the study of the stability of a static solution at a , the effective potential given by 0 we perform a Taylor expansion of V (a) about a0 until 4 4 4 2 2 2 2 256σ (a)π a 32π a σ (a)Ξ(a)+ζ (a) second–order terms, given by V (a)= − 2 2 2 , (66) − 64σ (a)π a 1 V (a) = V (a )+V ′(a )(a a )+ V ′′(a )(a a )2 where, for computational convenience, we introduce the 0 0 − 0 2 0 − 0 functions Ξ(a) and ζ(a) with the following definitions: +O[(a a )3], (70) − 0 2M q2 a2 Ξ(a) = 1 + + 1 , (67) where a prime denotes the derivative with respect to a. − a a2 −R2 Here we adapt and apply the criteria for stability anal- 2M q2 a2 ysis of a static configuration at a = a0, which requires ′ ζ(a) = 1 + 2 1 2 . (68) V (a0)= V (a0) = 0. The condition for stability is that − a a − −R ′′ V (a0)>0, to guarantee that the second derivative of the This allows us to write the potential in a second-order potential is positive. In order to determine the stability differential form as ′′ of the gravastar we first calculate V (a0). Using Eq. (55) ′ ′ and by introducing a new parameter η0 = (a0)/σ (a0) 1 P V ′′(a)= ∆ σ′′+16σ4π2Ξ′′a4 σ2ζa2ζ′′+∆ σ′ into Eq. (69), we obtain 32σ4a4π2 1 − 2 ′ 2 6 4 4 2 ′2 2 2 ′ 2 2 +∆3(σ ) 256σ π a σ ζ a +4σ ζ ζa 3σ ζ , − − − (69) 1 3 3 V ′′(a ) = 1024 η + π4R4a8σ6 1024σ5π4R4a8 η + p 0 32π2σ4a8R4 − 0 4 0 0 − 0 0 0 2 0 0 0 3 1 1 a4 2 64 16a4p2π2+Ma q2 R2+ a4 π2R2a4σ4 48p2 Ma q2 R2 0 − 0 0 0−2 2 0 0 0 − 0−2 − 2 1 1 3 13 q2 a4 +16p M η a q2 η + R2 1/2 η a4 Ma R2 0 σ 0−2 0−2 0 2 − 0− 2 0 0− 2 − 2 0 1 3 3 1 1 + 16M 2a2η 16Mq2 η a +4 η q4 R4 16 M η a q2 η a4R2 0 0− 0−4 0 0−4 − 0−4 0−2 0−4 0 15 +4 η a8 σ2 . (71) 0− 4 0 0 ′′ Therefore, by demanding that V (a0) > 0, we find an inequality for the η0 parameter, which yields 768σ6π4R4a8+1536σ5π4R4a8p +64π2R2a4σ4Θ +σ2Θ +8πσa Θ +48p2Θ η > 0 0 0 0 0 0 0 1 0 2 0 3 0 4 , (72) 0 4σ (σ +p )( 256σ4π4a8R4+4M 2a2R4 4Ma R4q2 4Ma5R2+R4q4+2a4R2q2+a8) 0 0 0 − 0 0 0 − 0 − 0 0 0 if σ (σ +p )( 256σ4π4a8R4+4M 2a2R4 4Ma R4q2 4Ma5R2+R4q4+2a4R2q2+a8)>0, 0 0 0 − 0 0 0 − 0 − 0 0 0 and 768σ6π4R4a8+1536σ5π4R4a8p +64π2R2a4σ4Θ +σ2Θ +8πσa Θ +48p2Θ η < 0 0 0 0 0 0 0 1 0 2 0 3 0 4 , (73) 0 4σ (σ +p )( 256σ4π4a8R4+4M 2a2R4 4Ma R4q2 4Ma5R2+R4q4+2a4R2q2+a8) 0 0 0 − 0 0 0 − 0 − 0 0 0 if σ (σ +p )( 256σ4π4a8R4+4M 2a2R4 4Ma R4q2 4Ma5R2+R4q4+2a4R2q2+a8)<0. 0 0 0 − 0 0 0 − 0 − 0 0 0 115101-8 Chinese Physics C Vol. 42, No. 11 (2018) 115101 For notational simplicity, in Eqs. (72) and (73), we use 3q2 a4 Θ = 16a4 p2 π2+Ma R2+ 0 , 1 0 0 0− 2 2 Θ = ( 4Ma q2+3q4)R4+2a4 (q2 6Ma )R2+15a8, 2 − 0 0 − 0 0 3q2 13a4 q2 a4 Θ = Ma + R2 0 Ma R2 0 , 3 0 2 − 2 0− 2 − 2 q2 a4 2 Θ = Ma R2 0 . 4 0− 2 − 2 Obviously, to ensure stability, we demand that in equilibrium the configurations satisfy the usual condition Fig. 5. (color online) Stability regions in terms of ′′ V (a0)>0. To justify our assumption for a given set of η0 as a function of a0 for M =2 and q = 2. The ′′ parameters and find the range of a0 for which V (a0)>0, shaded region is for a0 ever, η0 may lie outside the range of (0, 1] on the surface layer. For an extensive discussion see Refs.[21, 35]. 10 Stability analysis using the surface mass of the thin shell In this section we study the gravastar stability through the surface mass of the thin shell, following 2 Ref. [39], which is given by ms =4πa σ. For the stability analysis, we do not need to introduce a particular param- eter; rather, we can simply choose σ(a), or equivalently ms, as an arbitrarily specifiable function that encodes Fig. 3. (color online) Stability regions in terms of the whole gravastar stability. η0 as a function of a0 for M =1 and q = 1. The For our purpose, in the exterior Reissner-Nordstr¨om shaded region is for a0 2 2 2M q a0 ms(a0)= a0 1 + 2 1 2 , (74) − − a0 a0 − −R while for first–order differentiation we have 2a2 M 1 0 1 ′ − R2 − a0 ms(a0)= . (75) a2 − 2 1 0 2M q 2 1 + −R − a a2 0 0 For a static shell, one can derive the inequality for ′′ ms (a0), which may be used for a stable configuration, Fig. 4. (color online) Stability regions in terms of as the above expression contains two terms with oppo- η0 as a function of a0 for M =2 and q =1.5. The site signs. shaded region is for a 115101-9 Chinese Physics C Vol. 42, No. 11 (2018) 115101 analysis, we shall adapt the cases of σ>0, which corre- 11 Summary and discussion spond to positive surface energy densities (see the Ap- pendix for more details). By considering a stable static As an alternative to black holes, compact objects like solution at a0, we must have: gravastars have been proposed as a different final state 2 of a gravitational collapse, though the evidence for the a 2 a 2 M q 2 0 2 0 3 1 existence of black holes is well accepted astrophysically. ′′ R R − a0 M − In this paper, a spherically symmetric charged thin-shell a0ms (a0)> 3/2 3/2 a2 − 2M q2 gravastar has been investigated for a certain range of 1 0 1 + −R2 − a a2 parameters where the metric potentials and electromag- 0 0 (76) netic fields are related in some particular relation called Now, from the master equation (76), we mimic the stable Guilfoyle’s solutions. We consider a gravastar composed equilibrium regions of the respective solutions. To deter- of a de Sitter core, a thin shell, and an exterior Reissner- mine the stability regions of this solution, we choose the Nordstr¨om electrovacuum region. parameters such that the transition layer is located at The most relevant property is that within the δ-shell some value between q Appendix: static gravastars We derive the surface stress-energy tensor in terms of sur- 1 2M q2 a2 σ(a ) = − 1− + − 1− 0 face energy density σ and surface pressure P, around a stable 0 2 2 4πa0 "r a0 a r R # solution situated at a0. To see the qualitative behavior de- 1 x q2 y2 picted in Figs. 1 and 2, let us start with Eqs. (59, 60) and − − 2− − = 1 2x+ 2 x 1 2 , (A1) introduce new variables x = M/a0 and y = M/R. 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