Stability of Charged Thin-Shell Gravastars with Quintessence
aa r X i v : . [ g r- q c ] F e b Stability of Charged Thin-ShellGravastars with Quintessence
M. Sharif ∗ and Faisal Javed † Department of Mathematics, University of the Punjab,Quaid-e-Azam Campus, Lahore-54590, Pakistan.
Abstract
This paper develops a new solution of gravitational vacuum starin the background of charged Kiselev black holes as an exterior mani-fold. We explore physical features and stability of thin-shell gravastarswith radial perturbation. The matter thin layer located at thin-shellgreatly affects stable configuration of the developed structure. We as-sume three different choices of matter distribution such as barotropic,generalized Chaplygin gas and generalized phantomlike equation ofstate. The last two models depend on the shell radius, also known asvariable equation of state. For barotropic model, the structure of thin-shell gravastar is mostly unstable while it shows stable configurationfor such type of matter distribution with extraordinary quintessenceparameter. The resulting gravastar structure indicates stable behaviorfor generalized Chaplygin gas but unstable for generalized phantom-like model. It is also found that proper length, entropy and energywithin the shell show linear relation with thickness of the shell.
Keywords:
Gravastars; Israel formalism; Stability analysis.
PACS: ∗ [email protected] † [email protected] Introduction
The study of final outcomes of gravitational collapse is an interesting topicthat explores the formation of various compact objects such as white dwarfs,neutron stars, naked singularities and black holes (BHs). The collapse end-state is a widely accepted research field from many perspectives, both theo-retical and observational. The classical general relativity faces some majorscientific issues precisely related to the paradoxical characteristics of BHs andnaked singularities. An astronomical substance hypothesized as a substitutefor the BH is a gravastar (gravitational vacuum star) based on the idea ofMazur’s and Motola’s theory [1, 2]. The basic idea is to prevent the forma-tion of event horizons and singularities by stopping the collapse of matter ator near the position of event horizon. A gravastar appears similar to a blackhole but does not contain event horizon and singularity.Gravastars are of purely theoretical interest and can be described inthree different regions with the specific equation of state (EoS). The firstregion is referred to as an interior (0 ≤ r < r ), second is the intermediate( r < r < r ) and third is denoted as an exterior region ( r < r ). In the firstregion, the isotropic pressure ( p = − σ , where σ represents the energy den-sity) produces a repulsive force on the intermediate thin-shell. It is assumedthat the intermediate thin-shell is protected by ultrarelativistic plasma andfluid pressure ( p = σ ). The exterior region has zero pressure ( p = 0 = σ ) andcan be supported by the vacuum solution of the field equations. It contains astable thermodynamic solution and maximum entropy for small fluctuations.Visser’s cut and paste method provides a general formalism for the construc-tion of thin-shell from the joining of two different spacetimes at hypersurface[3]. Mazur and Mottola [2] considered this approach to construct thin-shellgravastar from the matching of exterior Schwarzschild BH with interior deSitter (DS) spacetime. This approach is very useful to avoid the presence ofevent horizon as well as central singularity in the geometry of gravastars.The matter surface at thin-shell creates a sufficient amount of pressure toovercome the force of gravity effects that help to maintain its stable configu-ration. For the description of Mazur-Mottola scenario, Visser and Wiltshire[4] introduced the simplest model from the matching of exterior and interiorgeometries through the cut and paste approach. They also analyzed stableconfiguration of the developed structure for suitable choice of EoS for thetransition layers. Carter [5] extended this concept by the joining of interiorDS spacetime and exterior Reissner-Nordstr¨om (RN) BH. They examined2he effects of EoS on the modeling of thin-shell gravastars. Horvat et al. [6]presented theoretical model of gravastars with electromagnetic field and stud-ied the role of charge on the stable configuration of gravastars. Rahaman etal. [7] studied physical features like proper length, entropy and energy con-tents of charged and charged free thin-shell gravastars in the backgroundof (2+1)-dimensional spacetime. They claimed that the presented solutionsare non-singular and physically viable as an alternative to BH. Banerjee etal. [8] investigated the braneworld thin-shell gravastars developed by usingbraneworld BH as an exterior manifold through cut and paste technique.Rocha and his collaborators [9] discussed stable configuration of thin-shell gravastars with perfect fluid distribution in Vaidya exterior spacetime.They proposed a dynamical model of prototype gravastars filled with phan-tom energy. It is found that the developed structure can be a BH, stable,unstable or “bounded excursion” gravastar for various matter distributionsat thin-shell. Horvat et al. [10] studied the geometry of gravastars withcontinuous pressure by using the conventional Chandrasekhar approach andderived EoS for the static case. Lobo and Garattini [11] investigated the sta-bility of noncommutative thin-shell gravastar and found that stable regionsmust exist near the expected position of the event horizon. ¨Ovg¨un et al. [12]developed thin-shell gravastar model from the matching of exterior chargednoncommutative BH with interior DS manifold. They found that the devel-oped structure follows the null energy condition and shows stable behavior forsome suitable values of physical parameter near the expected event horizon.Recently, we have developed regular thin-shell gravastars in the backgroundof Bardeen/Bardeen DS BHs as exterior manifolds through cut and pastemethod [13]. The stable configuration of the developed structure is exploredthrough radial perturbation. It is found that stable regions decrease for largevalues of charge and increase for higher values of the cosmological constant.The theoretical modeling of gravastar could be helpful for the better un-derstanding of dark energy role in the accelerated expanding behavior ofthe universe. Ghosh et al. [14] examined physical characteristics of gravas-tar model with Kuchowicz metric potential. They claimed that this modelovercomes the singularity problems that occurred for the geometry of BH ingeneral relativity. Shamir and Ahmad [15] investigated physical features ofgravastar model in the background of f ( G, T ) gravity. Yousaf et al. [16] ex-plored stable configuration of charged gravastar filled with isotropic fluid in f ( R, T ) gravity. They found linear relation among the physical features andthickness of the shell. Sharif and Waseem [17] discussed charged gravastars3ith conformal motion in f ( R, T ) gravity. There is a large body of literature[18]-[31] that explore the stable as well as dynamical configuration of thin-shell wormholes constructed from the matching various BHs with differentEoS.This paper presents the formalism of charged Kiselev thin-shell gravas-tars to explore stable configuration with different EoS. The paper has thefollowing format. Section explains the formalism of thin-shell gravastarsin the background of charged Kiselev BH. In section , we study the effectsof barotropic and variable EoS on the stable configuration of the developedstructure through radial perturbation. Finally, we summarize our results inthe last section. This section explores the geometrical construction of thin-shell gravstars fromthe joining of lower (Υ − ) and upper (Υ + ) manifolds through cut and pastetechnique. For this purpose, we consider DS spacetime as a lower manifoldand charged BH surrounded by the quintessence matter as an upper manifold.The motivation behind the consideration of this model can be explained asfollows. The matter with negative pressure can be characterized for thecurrent evolutionary phase of the universe with cosmological constant andquintessence [32]. The mathematical representation of quintessence matterdistribution that linear relates energy density ( σ q ) and pressure ( p q ) is p q = wσ q , where ω denotes the quintessence parameter. This parameter explainsthat the universe is in the phase of accelerated expansion if − < ω < − / ω > − / ω = − /
3. This means that observers must have future horizons inall accelerated models [33]. In an accelerated expanding universe, two objectsseparated with a relative fixed distance r must achieve relative speed to thespeed of light after some time and will no longer communicate. For the caseof decelerated expansion, the breakdown of such a communication does nothappen whereas it becomes less relativistic with time. However, the speed ofrelative moving observers must be constant for the case of ω = − /
3. Theycan communicate but cannot maintain this forever as they recede away fromeach other.Kiselev [34] introduced uncharged and charged BH surrounded by thequintessence matter distribution as a static spherically symmetric solution4f the field equations. The line element of charged Kiselev BH is given as ds = − Ψ( r + ) dt + Ψ( r + ) − dr + r dθ + r sin θ + dφ , (1)where Ψ( r + ) = 1 − mr + − αr ω +1+ + Q r ,m is the mass of BH, Q denotes the charge of BH, α stands for the Kiselevparameter and ω is the quintessence parameter with − < ω < − /
3. Theboundary values of EoS parameter recover the case of cosmological constant(extraordinary quintessence) for ω = − ω = 0 is referred to as thedust fluid. If Q = 0, then it reduces to Kiselev BH and RN BH is recov-ered when Kiselev parameter vanishes. The charged Kiselev BH reduces toSchwarzschild BH in the absence of both charge and Kiselev parameter. Thecorresponding metric function of Kiselev BH has the following formΨ( r + ) = 1 − mr + − αr ω +1+ . Extreme BHs are expected to have both stable and unstable proper-ties, this makes their analysis very interesting and challenging. We consider ω = − / ∈ ( − , − /
3) to observe the event horizon of Kiselev BH. Thecorresponding event horizons are given as r h = 1 ± √ − αm α . It is found that • for α = 1 / m , it denotes extreme Kiselev BH, • for α < / m , it represents the non-extreme Kiselev BH, • for α > / m , it shows naked singularity.Since the charged Kiselev BH metric function is much complicated than RNand Kiselev BH, so its event horizon for ω = − / • for Q = α (cid:0) − mα − − mα ) / (cid:1) , it denotes extreme chargedKiselev BH, 5 for Q > α (cid:0) − mα − − mα ) / (cid:1) , it represents the non-extreme charged Kiselev BH, • for Q < α (cid:0) − mα − − mα ) / (cid:1) , it shows naked singular-ity.The line element of DS geometry is given as ds − = − Φ( r − ) dt − + Φ − ( r − ) dr − + r − dθ − + r − sin θ − dφ − , (2)where Φ( r − ) = 1 − r − /β and β is a nonzero positive constant. We use cutand paste method to obtain the geometry of thin-shell gravastars from thematching of two distinct spacetimes Υ − and Υ + . These manifolds have themetric functions defined by g ± µν ( x µ ± ) with independent coordinates x µ ± andbounded by the hypersurfaces ∂ Υ ± with induced metrics h ± ij , respectively.According to the Darmoise junction conditions, the induced metrics are iso-metric and follow the relation h + ij ( ξ i ) = h ij ( ξ i ) = h − ij ( ξ i ), where ξ i representsthe coordinates of ∂ Υ ± . These geometries are glued at the hypersurface toobtain the single manifold Υ = Υ + ∪ Υ − with boundary ∂ Υ = ∂ Υ + = ∂ Υ − .Mathematically, these spacetimes can be described asΥ ± = { x µ ± | t ± ≥ T ± ( τ ) and r ≥ b ( τ ) } , where τ and b ( τ ) denote the proper time and radius of thin-shell. The cor-responding hypersurface that linked these geometries can be parameterizedas ∂ Υ = { ξ i | t ± ≥ T ± ( τ ) and r = b ( τ ) } . The induced 3D metric at hypersurface ( h ij ) can be expressed as ds ∂ Υ = h ij dξ i dξ j = − dτ + b ( τ ) dθ + b ( τ ) sin θdφ , where ξ i = ( τ, θ, φ ). The normal vector components of g µν on the ∂ Υ aredefined as n µ = f ( r, b ( τ )) ,µ | f ( r, b ( τ )) ,ν f ( r, b ( τ )) ,ν | / , where f ( r, b ( τ )) = r − b ( τ ) = 0 represents the function of ∂ Υ and b ( τ ) = b denotes the shell’s radius. The components of normal vectors correspondingto upper and lower spacetimes are n µ + = ˙ b − mb − αb ω +1 + Q b , r − mb − αb ω +1 + Q b + ˙ b , , ! , (3)6 µ − = ˙ b − b β , s − b β + ˙ b , , ! , (4)respectively. Here, dot represents derivative with respect to τ . The normalvector satisfies the condition n µ n µ = 1 for spherical symmetric manifolds.The discontinuity in the second fundamental form (extrinsic curvature) ex-ist due to the presence of matter surface at ∂ Υ. The extrinsic curvaturecomponents for both geometries are K τ + τ = α (3 ω + 1) b − ω + 2 bm − Q + 2¨ bb b q − mb − αb ω +1 + Q b + ˙ b , (5) K θ + θ = 1 b r − mb − αb ω +1 + Q b + ˙ b , (6) K τ − τ = − bβ + 2¨ b q − b β + ˙ b , (7) K θ − θ = 1 b s − b β + ˙ b , (8) K φ ± φ = sin θK θ ± θ , (9)The matter surface at thin-shell produces discontinuity in the extrinsiccurvatures of both spacetimes. If K + ij − K − ij = 0, then it represents thepresence of matter thin layer on ∂ Υ. The components of energy-momentumtensor ( S ij ) of such a matter surface are determined by the Lanczos equations.Mathematically, it can be expressed as S ij = − π { [ K ij ] − δ ij K } , (10)where [ K ij ] = K + ij − K − ij and K = tr [ K ij ] = [ K ij ]. The above equation interms of perfect fluid distribution becomes S ij = v i v j ( p + σ ) + pδ ij , (11)here v i denotes thin-shell velocity components. By considering Eqs.(5)-(11),we obtain σ and p in the following form σ = − πb (r − mb − αb ω +1 + Q b + ˙ b − s − b β + ˙ b ) , (12)7 = 2˙ b + 2 b ¨ b + α (3 ω − b − ω − mb + 28 πb q − mb − αb ω +1 + Q b + ˙ b − b + 2 b ¨ b + 2 − b β πb q − b β + ˙ b . (13)Here, we assume that ˙ b = ¨ b = 0, where b is the position of equilibriumshell’s radius. This shows that shell’s motion along the radial direction van-ishes at b = b . The respective expressions for surface stresses at b = b yield σ ( b ) = σ = − πb s − mb − αb ω +10 + Q b − s − b β , (14) p ( b ) = p = 18 πb α (3 ω − b − ω − m + 2 b b q − mb − αb ω +10 + Q b − β − b β q − b β . (15)The continuity of perfect fluid gives the relationship between the surfacestresses of thin-shell gravastars as4 π ddτ ( b σ ) + 4 πp db dτ = 0 , (16)which can be expressed as dσdb = − b ( σ + p ) . (17)The second order derivative of σ with respect to b yields d σdb = 2( p + σ ) b (cid:0) ς (cid:1) , (18)where ς = dp/dσ denotes the EoS parameter. Equations (16)-(18) are veryuseful to explore the dynamics and stable configurations of constructed ge-ometry with different types of matter distribution.For the physical viability of a geometrical structure, some constraintsmust be imposed known as energy conditions. The well-known energy con-ditions are null: σ + p >
0; weak: σ > σ + p >
0; strong: σ + 3 p > σ + p >
0; dominant: σ > σ ± p >
0. If these energy conditions areverified then the developed model is physically viable. Here, we are interested8igure 1: Plots of the null energy condition for charged Kiselev thin-shellgravatars. We examine graphical behavior of σ + p at b = b verses ( b , Q )(left plot) and ( b , m ) (right plot) for ω = − / α = 0 . β = 0 . A ). This section studies stability of thin-shell gravastars using linear perturbationin the radial direction at b = b with different variable EoS. The stable andunstable configurations of thin-shell gravastars can be analyzed through thebehavior of effective potential of thin-shell. The equation of motion of thin-shell that explains the stable as well as dynamical characteristics of respective9eometry is obtained directly from Eq.(12) as˙ b + Ω( b ) = 0 , (19)here Ω( b ) denotes the potential function of thin-shell gravastar asΩ( b ) = − ξ ( b ) π b σ + 12 ζ ( b ) − π b σ , (20)where ξ ( b ) = αb − ω − b β + 2 bm − Q b , ζ ( b ) = 2 + − αb − ω − b β − bm + Q b . The stable behavior of thin-shell gravastars is studied by using secondderivative of the effective potential at b = b . The basic conditions for thestable behavior can be written as Ω( b ) = 0 = Ω ′ ( b ) and Ω ′′ ( b ) > ′′ ( b ) <
0, then it shows unstable behavior and it is unpredictable ifΩ ′′ ( b ) = 0 [26]. To check the stability through radial perturbation, we lin-earize the potential function using Taylor series expansion around equilibriumradius b as followsΩ( b ) = Ω( b ) + Ω ′ ( b )( b − b ) + 12 Ω ′′ ( b )( b − b ) + O [( b − b ) ] . We examine that Ω( b ) = 0 = Ω ′ ( b ), hence it reduces toΩ( b ) = 12 ( b − b ) Ω ′′ ( b ) . (21)The corresponding second derivative of Ω( b ) at b = b becomesΩ ′′ ( b ) = 2 M ( b ) M ′ ( b ) b − b ξ ( b ) ξ ′′ ( b )2 M ( b ) + 2 b ξ ( b ) M ′ ( b ) ξ ′ ( b ) M ( b ) − M ′ ( b ) b − b ξ ′ ( b ) M ( b ) − b ξ ( b ) M ′ ( b ) M ( b ) − b ξ ( b ) ξ ′ ( b ) M ( b ) + ζ ′′ ( b )2 − M ( b ) b + b ξ ( b ) M ′′ ( b )2 M ( b ) − M ( b ) M ′′ ( b )2 b − ξ ( b ) M ( b ) + 2 b ξ ( b ) M ′ ( b ) M ( b ) , (22)10here M ( b ) = 4 πb σ denotes the total mass distribution at equilibriumshell’s radius. The corresponding first and second derivatives of the totalmass with respect to b at b = b become M ′ ( b ) = − πb p , M ′′ ( b ) = − πp + 16 πς ( σ + p ) , and ς = dp/dσ | b = b .Firstly, we begin with barotropic EoS to discuss the stability of the de-veloped geometry. It gives linear relation between the surface stresses ofthin-shell as p = γσ with real constant γ . Consequently, the solution ofconservation equation (17) for barotropic EoS is given as σ = (cid:0) b b − (cid:1) γ ) σ . (23)The corresponding potential function becomesΩ( b ) = ζ ( b )2 − ξ ( b ) (cid:0) b b (cid:1) − γ +1) π b σ − π b σ (cid:18) b b (cid:19) γ +1) , (24)which turns out to be zero at throat radius b = b . The corresponding firstderivative of Ω( b ) yieldsΩ ′ ( b ) = ζ ′ ( b )2 − ξ ( b ) (2 γξ ( b ) + b ξ ′ ( b ) + ξ ( b ))32 π σ b + 8 π σ (2 γ + 1) b , (25)which vanishes only if γ = − π σ b − π σ b ζ ′ ( b ) + b ξ ( b ) ξ ′ ( b ) + ξ ( b ) π σ b − ξ ( b ) ) . (26)The second derivative of Ω( b ) at b = b yieldsΩ ′′ ( b ) = ζ ′′ ( b )2 − π σ b { b ξ ( b ) ((8 γ + 4) ξ ′ ( b ) + b ξ ′′ ( b ))+ (cid:0) γ + 6 γ + 1 (cid:1) ξ ( b ) + b ξ ′ ( b ) (cid:9) − π σ (2 γ + 1)(4 γ + 3) , (27)This equation explains stable and unstable configurations of thin-shell gravas-tars for barotropic EoS. Due to complexity of this expression, we use numer-ical approach to observe the effects of physical parameters on the stability11
200 400 600 800 1000 - - - - b W '' H b L Α= Ω=- (cid:144) b W '' H b L Α= Ω=- Figure 2: Stability of thin-shell gravastars with barotropic EoS for β = 0 . m = Q with different values of ω . The left plot shows unstable behavior andright plot expresses the stable structure.of developed structure. We study the graphical behavior of Ω ′′ ( b ) by usingEqs.(26) and (14). It is found that stable structure of thin-shell is greatlyaffected by the presence of quintessence EoS parameter. We examine thatthin-shell expresses unstable behavior for every values of Q , m , α and β with ω = − / ω except for extraordinary quintessence parameter ω = − ω = − pσ = η , where η <
0. A remarkable property of the Chaplygin gas is thatthe squared sound velocity v s = η/σ is always positive even in the case ofexotic matter. Varela [18] considered the EoS of the type p = p ( σ, b ) to dis-cuss the stability of thin-shell wormhole developed from two equivalent copiesof the Schwarzschild BH. Such type of EoS is known as variable EoS. Thegeneralized form of the Chaplygin gas presents the mathematical formulationin which surface pressure depends on the radius of the shell.Therefore, we consider general form of Chaplygin EoS ( p = p ( σ, b )) to12tudy the stable behavior of the respective geometry, i.e., p = b n ησ with realconstants η < n [18]. It is observed that the Chaplygin gas model isrecovered for n = 0 [38]. The respective solution of conservation equation forsuch a model can be written as σ = ( n − σ b n +40 b n + 4 ηb b n − ηb n b b n +4 b n ( n − . (28)The effective potential for this model turns out to beΩ( b ) = − π b − n − b − n ( b b n (( n − σ b n − η ) + 4 b ηb n ) n − ζ ( b )2 − ( n − b n +2 ξ ( b ) b n π ( b b n (( n − σ b n − η ) + 4 b ηb n ) . (29)It is observed that Ω( b ) = 0 and Ω ′ ( b ) becomesΩ ′ ( b ) = − ηb − n − ξ ( b ) π σ + 16 π ηb − n − ξ ( b ) ξ ′ ( b )32 π σ b − ξ ( b ) π σ b + 8 π σ b + ζ ′ ( b )2 . For Ω ′ ( b ) = 0, we have η = − σ b n (256 π σ b + 16 π σ b ζ ′ ( b ) − b ξ ( b ) ξ ′ ( b ) − ξ ( b ) )2 (256 π σ b − ξ ( b ) ) . (30)Consequently, Ω ′′ ( b ) has the following formΩ ′′ ( b ) = − η b − n +2)0 ξ ( b ) π σ − ηb n − n +2)+10 ξ ( b ) ξ ′ ( b )4 π σ + ηnb n − n +2)0 ξ ( b ) π σ − ηb n − n +2)0 ξ ( b ) π σ − π ηb n − n +2)+40 − π ηnb n − n +2)+40 − b n − n +2)+20 ξ ( b ) ξ ′′ ( b )32 π σ − b n − n +2)+10 ξ ( b ) ξ ′ ( b )8 π σ − b n − n +2)+20 ξ ′ ( b ) π σ − b n − n +2)0 ξ ( b ) π σ − π σ b n − n +2)+40 + 12 b n − n +2)+40 ζ ′′ ( b ) . (31)13
200 400 600 800 10000.0010.0020.0030.0040.0050.0060.0070.008 b W '' H b L Α= Ω=- (cid:144)
3, n = b W '' H b L Α= Ω=- = Figure 3: Stable behavior of thin-shell gravastars with Chaplygin gas model( n = 0) for β = 0 . m = Q with different values of ω .Now, we observe the effects of the generalized Chaplygin gas EoS on thestability of developed geometry. In this regard, we observe the graphicalbehavior of Ω ′′ ( b ) for this model. It is found that thin-shell expresses stablebehavior for every choice of the physical parameters except ω = − n = 0 (Figure 3). This shows that thin-shell becomes stable for the choiceof Chaplygin gas model ( n = 0) and represents unstable behavior only for ω = − n = 0) shows stable behavior for every choice of ω with n = 1 (right plot of Figure 4). We see that stable behavior (Ω ′′ ( b ) > n as shown in Figure 5.Finally, we study the effects of generalized phantomlike variable EoS onthe stability of thin-shell [18] whose EoS is p = Θ σb n with real constants Θ and n . The phantomlike EoS is recovered if n = 0 [39]. By using this expressionin Eq.(17), we have σ = b b − σ e Θ ( b − n − b − n ) n , (32)and it follows thatΩ( b ) = − b ξ ( b ) e ( b − n − b − n ) n π σ b − π σ b e ( b − n − b − n ) n b + ζ ( b )2 . (33)It is noted that Ω( b ) = 0 and by considering Ω ′ ( b ) = 0, we obtainΘ = − b n (256 π σ b + 16 π σ b ζ ′ ( b ) − b ξ ( b ) ξ ′ ( b ) − ξ ( b ) )256 π σ b − ξ ( b ) , (34)14
200 400 600 800 1000 - - - - - b W '' H b L Α= Ω=-
1, n = b W '' H b L Α= Ω=-
1, n = Figure 4: Stability of thin-shell gravastars with generalized Chaplygin gasEoS with different values of n . For ω = −
1, the left plot shows unstablebehavior for n = 0 and right plot expresses the stable structure for n = 1. b W '' H b L Α= Ω=- (cid:144)
3, n = b W '' H b L Α= Ω=- (cid:144)
3, n = Figure 5: Stable behavior of thin-shell gravastars for different values of n .The stability of developed structure is enhanced for large values of n .15nd henceΩ ′′ ( b ) = − Θ b − n − ξ ( b ) π σ − π Θ σ b − n − Θ b − n − ξ ( b ) ξ ′ ( b )8 π σ − Θ b − n − ξ ( b ) π σ − Θ(1 − n ) b − n − ξ ( b ) π σ − π Θ σ b − n + 8 π Θ( − n − σ b − n − ξ ( b ) ξ ′′ ( b )32 π σ b − ξ ( b ) ξ ′ ( b )8 π σ b − ξ ′ ( b ) π σ b − ξ ( b ) π σ b − π σ + ζ ′′ ( b )2 . (35)For the general form of phantomlike EoS, we see that thin-shell shows ini-tially stable behavior then expresses unstable configuration for every choiceof physical parameters (Figures 6 and 7). We conclude that the constructedgeometry is neither stable nor unstable completely for the choice of bothphantomlike and general form of phantomlike EoS. This paper investigates the construction of thin-shell gravastars from thematching of two different spacetimes, i.e., DS as a lower spacetime andcharged Kiselev BH as an upper manifold. These geometries are connectedthrough the well-known cut and paste method. We match these manifolds at r = b with b > r h to avoid the presence of event horizon ( r h ) and singularityin the developed structure. The presence of matter thin layer at the joiningsurface produces discontinuity in the extrinsic curvature. It is found that thenull energy condition is verified for the developed structure (Figure 1). Wehave studied stable characteristics of thin-shell gravastars with barotropictype fluid distribution and two variable EoS, i.e., generalized Chaplygin gasand phantomlike EoS.For barotropic model, we have obtained stable solution for the choice of ω = − ω (Figure 2). It isinteresting to mention here that this model mostly indicates unstable behav-ior for thin-shell WHs in several spacetimes [18, 19, 30, 31]. These resultsexpress that the stable solution can be obtained through barotropic modelfor some suitable choice of physical parameters. The stable structure is ob-tained for Chaplygin gas model ( n = 0) for every choice of ω other than16 .20 0.25 0.30 0.35 0.40 -
20 000 -
10 000010 00020 00030 00040 000 b W '' H b L Α= Ω=- (cid:144)
3, n = -
20 000 -
10 000010 00020 00030 000 b W '' H b L Α= Ω=-
1, n = Figure 6: Stable and unstable behavior of thin-shell gravastars with phan-tomlike EoS ( n = 0) for different values of ω . It shows stable behaviorinitially then expresses unstable configuration for every choice of ω . -
100 000 -
50 000050 000100 000150 000 b W '' H b L Α= Ω=- (cid:144)
3, n = -
40 000 -
20 000020 00040 00060 000 b W '' H b L Α= Ω=-
1, n = Figure 7: Stable and unstable behavior of thin-shell gravastars for generalizedphantomlike EoS with different values of ω .17xtraordinary quintessence parameter ω = −
1. For generalized Chaplygingas EoS, we have obtained stable solution for every values of physical pa-rameter and found more stable structure for higher values of n (Figures 3,4 and 5. Finally, for generalized phantomlike EoS, thin-shell shows initiallystable behavior and then expresses unstable configuration for every choice ofthe physical parameters (Figures 6 and 7).We conclude that charged Kiselev thin-shell gravastars are more stable forthe choice of generalized Chaplygin gas model. It is worthwhile to mentionhere that this model is more stable with considered EoS than thin-shell WHsin the background of various BHs [18, 19, 30, 31]. This shows completelystable structure of thin-shell gravastar with extraordinary quintessence pa-rameter for both barotropic and generalized Chaplygin gas model. Appendix A
We also explore some physical features of the developed structure, i.e., properlength, entropy and energy contents within the shell’s region. Since theconstructed geometry is the matching of two different spacetimes, so the stiffperfect fluid moves along these spacetimes through the shell region. Thelower and upper boundaries of the shell are r = b and r = b + ǫ , respectively.The proper thickness of the shell is denoted by ε which is a very small positivereal number (0 < ε ≪ l = Z b + ǫb p Ψ − ( r ) dr = Z b + ǫb dr q − mr − αr ω +1 + Q r . (36)This integral cannot be solved analytically due to the complicated expressionof Ψ( r ). Therefore, we solve it by assuming q Ψ − ( r ) = dj ( r ) dr as l = Z b + ǫb dj ( r ) dr dr = j ( b + ǫ ) − j ( b ) ≈ ǫ dj ( r ) dr | r = b = ǫ q Ψ − ( b ) , (37)where ǫ ≪ l = ǫ (cid:20) − mb − αb ω +1 + Q b (cid:21) − . (38)18 =0.5Q = Ε S Α= = Ω=- (cid:144) m=0.2m=0.3m=0.5 Ε S Α= = Ω=- (cid:144) Figure 8: Behavior of entropy versus thickness of the shell with β = 0 . b = 1.It is noted that the proper length of the shell clearly depends on the chargeas well as the mass of the BH. Equation (38) shows that the proper lengthand thickness of the shell are proportional. It is found that the length ofthe shell decreases by an increasing charge of the geometry and increases byincreasing the mass of the BH.Entropy is related to the measure of disorderness or disturbance in ageometrical structure. We study the entropy of thin-shell gravastars thatexplains the disorderness in the geometry of gravastar. According to thetheory of Mazur and Mottola, charged gravastar has zero entropy density forthe interior region. Using the concept of Mazur and Mottola, we evaluatethe entropy of thin-shell gravastar through the expression [14] S = Z b + ǫb πr h ( r ) p Ψ − ( r ) dr. (39)The entropy density for local temperature can be expressed as h ( r ) = ϑK B ~ r p ( r )2 π , (40)where ϑ is a dimensionless parameter. Here, we take Planck units ( K B =1 = ~ ) so that the shell’s entropy becomes [14] S = ǫϑb p πp ( b )Ψ − ( b ) . (41)It is shown that entropy of the shell’s region is also proportional to the shell’sthickness. We use this equation to examine the contribution of charge and19 =0.5Q = Ε ¶ Α= Ω=- (cid:144)
3, m = m=0.2m=0.3m=0.5 Ε ¶ Α= Ω=- (cid:144)
3, Q = Figure 9: Behavior of the energy within the shell verses thickness of the shellwith β = 0 . b = 1.mass of BH on the entropy of shell graphically. Figure 8 shows the linearrelation between entropy and thickness for different values of the physicalparameter. It is found that the entropy of shell region increases by increasing Q and decreases for large values of m . The interior region of gravastars obeysthe EoS p = − σ which represents negative energy zone with non-attractiveforce. The energy distribution in the shell’s region can be determined as [14] ε = Z b + ǫb πr σ ( r ) dr ≈ ǫπb σ ( b ) . (42)The energy contents depend on the thickness of the shell, mass and charge ofthe geometry. We see that energy within the shell decreases for large valuesof charge and increases for large values of mass as shown in Figure 9.It is concluded that these features are proportional to the thickness ofthe shell and are greatly affected by the charge and mass of the BH which isconsistent with the literature [14]-[16]. Acknowledgement
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