aa r X i v : . [ phy s i c s . g e n - ph ] A ug Charged Gravastars in Rastall-Rainbow Gravity
Ujjal Debnath ∗ Department of Mathematics,Indian Institute of Engineering Science and Technology,Shibpur, Howrah-711 103, India.
In this work, we have considered the spherically symmetric stellar system in the contexts ofRastall-Rainbow gravity theory in presence of isotropic fluid source with electro-magnetic field. TheEinstein-Maxwell’s field equations have been written in the framework of Rastall-Rainbow gravity.Next we have discussed the geometry of charged gravastar model. The gravastar consists of threeregions: interior region, thin shell region and exterior region. In the interior region, the gravastarfollows the equation of sate (EoS) p = − ρ and we have found the solutions of all physical quantitieslike energy density, pressure, electric field, charge density, gravitational mass and metric coefficients.In the exterior region, we have obtained the exterior Riessner-Nordstrom solution for vacuum model( p = ρ = 0). Since in the shell region, the fluid source follows the EoS p = ρ (ultra-stiff fluid) andthe thickness of the shell of the gravastar is infinitesimal, so by the approximation h ( ≡ A − ) ≪ ǫ ) due to the approximation ( ǫ ≪ I. INTRODUCTION A gravastar is astronomically hypothetical condensedobject which is a gravitationally dark cold vacuumcompact star or gravitational vacuum condensate star.Mazur and Mottola [1, 2] have established the gravastarsolution in the concept of Bose-Einstein condensation togravitational systems. The gravastar is singularity freeobject which is spherically symmetric as well as supercompact. It has also the property that it has no eventhorizon. So the gravastar is a substitute of black holei.e., the existence of compact stars minus event horizons.The gravastar consists of three regions: (i) Interior region (0 ≤ r < r ), (ii) Shell region ( r < r < r ) and (iii) Exterior region ( r < r ), where r and r are innerand outer radii ( r < r ). In the interior region, theisotropic pressure produces a force of repulsion over theintermediate thin shell. So the equation of state (EOS)of the fluid satisfies p = − ρ which describes the de-Sitterspacetime. The intermediate thin shell region consistsof ultra-stiff perfect fluid satisfies the EoS p = ρ . Theexterior region consists of vacuum with EOS p = ρ = 0which is described by the Schwarzschild solution. Visser[3] developed the mathematical model of the gravastarand described the stability of gravastar by taking somerealistic values of EoS parameter.DeBenedictis et al [4] have found the gravastar solutionsby taking continuous pressures and the equation of state.The anisotropic pressure for the structure of gravastar has ∗ [email protected] been considered by Cattoen et al [5]. Bilic et al [6] havefound the gravastar solution in presence of Born-infeldphantom model. Carter [7] studied the stability of thegravastar. The Gravastar solutions in the framework ofconformal motion have been investigated by some authors[8–10]. Gravastar model in higher dimensional spacetimehas been discussed in refs [9, 11–13]. Several authors[14–20] have discussed the stable nature of gravastars.Charged gravastar models with its physical features havebeen analyzed in the works [8, 9, 12, 21–28]. Ray et al[29] have described the charged strange quark star modelin the framework of electromagnetic mass with conformalkilling vector.In the framework of modified gravity theory, lot ofworks on compact star, neutron star, strange star andgravastar have been found in the literature. Boehmer etal [30] have examined the existence of relativistic starsin f ( T ) modified gravity. The structure of neutron starsin modified f ( T ) gravity has been studied by Delidumanet al [31]. In refs [32, 33], the authors have studied theanisotropic strange stars in f ( T ) gravity model. Thestructures of relativistic stars in f ( T ) gravity and itsTolman-Oppenheimer-Volkoff (TOV) equations have beencomputed in [34]. In f ( T ) gravity model, the compactstar models have been studied in [35] and the neutronstar models have been discussed in [36]. Using the Kroriand Barua (KB) metric [37], the anisotropic compactstar models in GR, f ( R ), f ( G ) and f ( T ) theories havebeen studied in refs [38–41]. The gravastar solution in f ( R, T ) gravity model has been studied in [42, 43]. Alsothe Gravastar solution in f ( G, T ) gravity model has beenfound in [44].Rastall gravity theory is proposed by Rastall [45], whichis one of the alternative of modified gravity theory bymodification the Einstein’s general relativity. NeutronStars in Rastall Gravity have been obtained by Oliveiraet al [46]. A model of quintessence compact stars in theRastall’s theory of gravity has been obtained by Abhaset al [47]. Isotropic compact star model in Rastall theoryadmitting conformal motion has also been obtained in[48]. Anisotropic compact star model in the Rastall theoryof gravity has been discussed in [49]. Gravity’s rainbow[50] is a distortion of space-time which is an extensionof the doubly special relativity for curved space-times.The properties of neutron stars and dynamical stabilityconditions in the modified TOV in gravity’s rainbowhave been investigated by Hendi et al [51]. The gravity’srainbow and compact star models have been studied in[52]. The rainbow’s star models have also been studied in[53]. Recently, Mota et al [54] have studied the neutronstar model in the framework of Rastall-Rainbow theoriesof gravity.The main motivation of the work is to study the gravas-tar system in the framework of Rastall-Rainbow gravitywith the isotropic fluid and electromagnetic source andexamine the nature of physical parameters and stability ofthe gravastar. The organization of the work is as follows:In section II, we present the Rastall-Rainbow gravitytheory with electromagnetic field. Here we write theEinstein-Maxwell field equations for spherically symmetricsteller metric in the framework of Rastall-Rainbow gravity.Section III deals with the geometry of gravastar and wecompute the solutions in the three regions of gravastarmodel. In section IV, we analyze the physical aspects ofthe parameters of gravastar model. In section V, we inves-tigate the matching between interior and exterior regions.Due to the junction conditions, we compute the equationof state, mass of the thin shell region and examine thestability of the gravastar. Finally some physical analysisand fruitful conclusions of the work are drawn in section VI. II. RASTALL-RAINBOW GRAVITY
In Einstein’s General Relativity (GR), the conserva-tion law of energy-momentum tensor is T νµ ; ν = 0. TheRastall’s Gravity is a generalization of General Relativity,where Rastall [45] proposed the modification of conserva-tion law of energy-momentum tensor in curved space-timeand which is given by [54] T νµ ; ν = ¯ λR ,µ (1)where ¯ λ = − λ πG with λ is a constant called Rastall param-eter, which measures the deviation from GR and describesthe affinity of the matter field to couple with geometry. For λ = 1, the usual conservation law can be restored. Also forflat space-time, the Ricci scalar R = 0 and we may alsothe usual conservation law. So for the effect of Rastall’sgravity, λ = 1 and the space-time must be non-flat. The above equation can be written as (cid:0) T νµ − ¯ λδ νµ R (cid:1) ; ν = 0 (2)So for Rastall’s gravity, the Einstein’s equation can bemodified to the form R νµ − δ νµ R = 8 πG (cid:0) T νµ − ¯ λδ νµ R (cid:1) (3)which can be simplified to the form R νµ − λ δ νµ R = 8 πGT νµ (4)Now the trace of the energy-momentum tensor is given by T = (1 − λ ) R πG (5)So the above Einstein’s equation in Rastall’s gravity canbe written as R νµ − δ νµ R = 8 πG (cid:18) T νµ − (1 − λ )2(1 − λ ) δ νµ T (cid:19) (6)Now assume that the fluid source is composed of normalmatter and electro-magnetic field. So the energy momen-tum tensor can be written as T µν = T Mµν + T EMµν (7)where the energy-momentum tensor for normal matter isgiven by T Mµν = ( ρ + p ) u µ u ν + pg µν (8)where u µ is the fluid four-velocity satisfying u µ u ν = − ρ and p are the energy density and pressure of fluid. Further,the energy momentum tensor for electromagnetic field isgiven by [55] T EMµν = − π ( g δω F µδ F ων − g µν F δω F δω ) (9)where F µν is the Maxwell field tensor defined as in theform: F µν = Φ ν,µ − Φ µ,ν (10)and Φ µ is the four potential. The corresponding equationsfor Maxwell’s electromagnetic field are given by( √− g F µν ) , ν = 4 πJ µ √− g , F [ µν,δ ] = 0 (11)where J µ is the current four-vector satisfying J µ = σu µ ,the parameter σ is the charge density.Magueijo and Smolin [50] have proposed gravity’s Rain-bow, which is an extension of the doubly special relativityfor curved space-times. Gravity’s rainbow is a distortionof space-time induced by two arbitrary functions Π( x ) andΣ( x ) (called the Rainbow functions) satisfying E Π ( x ) − υ Σ ( x ) = m (12)where x = E / E P l . Here E , υ , m and E P l = √ ¯ hc /G arethe energy, momentum, mass of a test particle and Planckenergy respectively. Awad et al [56] and Khodadi et al [57]have chosen Π( x ) = 1 and Σ( x ) = √ x to study thesolutions corresponding to a nonsingular universe. Also tostudy of gamma ray burst, the exponential form of rainbowhas been applied in [56, 58]. In the absence of the testparticles, the Rainbow functions are satisfyinglim x → Π( x ) = 1 , lim x → Σ( x ) = 1 (13)Mota et al [54] have merged the Rastall’s gravity withRainbow’s gravity and applied the Rastall-Rainbow gravitytheory in the neutron star formation. We consider thespherically symmetric metric describing the interior space-time of a star in Rainbow gravity as [54] ds = − B ( r )Π ( x ) dt + A ( r )Σ ( x ) dr + r Σ ( x ) ( dθ + sin θdφ )(14)where A ( r ) and B ( r ) are functions of r . Since Π( x )and Σ( x ) depend on x = E / E P l and E is independent of r , so Π( x ) and Σ( x ) are independent of r . The metriccoefficients depend of the energy of the test particle. Sothe geometry of the space-time becomes energy dependent.It should be noted that the metric coordinates do notdepend on the energy of the particle.For the charged fluid source with density ρ ( r ), pressure p ( r ) and electromagnetic field E ( r ), the Einstein-Maxwell(EM) equations in the Rastall-Rainbow gravity can bewritten in the form [54] A ′ rA − r A + 1 r = 8 πG ¯ ρ , (15) B ′ rAB + 1 r A − r = 8 πG ¯ p , (16)and B ′′ AB − A ′ B ′ A B − B ′ AB − A ′ rA + B ′ rAB = 8 πG ¯ p (17)where¯ ρ = 1Σ ( x ) (cid:18) α ρ + 3 α p + 18 π ( α − α ) E (cid:19) , (18)¯ p = 1Σ ( x ) (cid:18) α ρ + (1 − α ) p + 18 π (4 α − E (cid:19) , (19)and¯ p = 1Σ ( x ) (cid:18) α ρ + (1 − α ) p + 18 π (1 − α ) E (cid:19) (20)with α = 3 λ − λ − , α = λ − λ −
1) (21) We observe that α + α = 1 and λ = 1 /
2. Adding equa-tions (15) and (16), we obtain1 rA (cid:18) A ′ A + B ′ B (cid:19) = 8 πG (¯ ρ + ¯ p ) = 8 πG Σ ( x ) ( ρ + p ) (22)From equation (15), we obtain [54] A ( r ) = (cid:18) − GM ( r ) r (cid:19) − (23)where the gravitational mass is M ( r ) = Z r πr ¯ ρ ( r ) dr (24)From the modification of conservation law of energy-momentum tensor, we obtain p ′ + ( ρ + p ) B ′ B = 18 πr ( r E ) ′ (25)and the electric field E is as follows E ( r ) = 1 r Z r πr σ ( r ) p A ( r )Σ( x ) dr (26)The term σ √ A Σ( x ) inside the above integral is known as thevolume charge density. III. GRAVASTAR
Here, we will derive the solutions of the field equa-tions for charged gravastar (charge is generated by electro-magnetic field) and analyze its physical as well as geo-metrical interpretations. Since there are three regions ofthe gravastar, so the geometrical regions of the gravastarhaving a finite extremely thin width within the regions D = r < r < r = D + ǫ where r and r are radii ofthe interior region and exterior region of the gravastar and ǫ is very small positive quantity. The three regions arestructured as follows: (i) Interior region R : 0 ≤ r < r with the equation of state (EOS) follows p = − ρ , (ii) Shellregion R : r < r < r with EOS follows p = ρ and (iii) Exterior region R : r < r with EOS follows p = ρ = 0. A. Interior Region
The interior region R (0 ≤ r < r = D ) of the gravastarfollows the EoS p = − ρ . From equation (17), we obtain therelation B ( r ) = kA − ( r ) (27)where k is constant >
0. There are 4 equations and 5unknown functions
A, B, ρ, p, E in the system. So onefunction is free to us. Here we may consider E is freefunction of r and is chosen by E ( r ) = E r m (28)where m is positive constant and E is function of x .Using equation (25), we obtain p = − ρ = k r m − k (29)where k = ( m +2) E πm and k is positive constant. From(24), we obtain M ( r ) = 12(2 λ − ( x ) (cid:18) πk r − E m (2 m + 3) r m +3 (cid:19) (30)From equation (23), we obtain the solution B ( r ) = kA − ( r ) = k (cid:20) − πGk λ − ( x ) r + 2 GE m (2 m + 3)(2 λ − ( x ) r m +2 (cid:21) (31)So the metric becomes (choose k = 1) ds = − ( x ) (cid:20) − πGk λ − ( x ) r + 2 GE m (2 m + 3)(2 λ − ( x ) r m +2 (cid:21) dt + 1Σ ( x ) (cid:20) − πGk λ − ( x ) r + 2 GE m (2 m + 3)(2 λ − ( x ) r m +2 (cid:21) − dr + r Σ ( x ) ( dθ + sin θdφ ) (32)The charge density for electric field is obtained in the form: σ ( r ) = ( m + 2) E r m − π (cid:20) − πGk λ − ( x ) r + 2 GE m (2 m + 3)(2 λ − ( x ) r m +2 (cid:21) (33)Also the gravitational mass of the interior region of thecharged gravastar can be found as M ( D ) = 12(2 λ − ( x ) (cid:18) πk D − E m (2 m + 3) D m +3 (cid:19) (34)We observe that the quantities A ( r ) , B ( r ) , M ( r ) , σ ( r )depend on the Rastall parameter λ and Rainbow functionΣ( x ). B. Shell Region
In the shell region R ( D = r < r < r = D + ǫ ), weassume that the thin shell region contains stiff perfect fluidwhich obeys EoS p = ρ . For this EoS, it is very difficult toobtain the solution from the field equations. So we shallassume the limit 0 < A − ≡ h ≪ h ≈ p = ρ , the field equations (15) - (17) reduce to thefollowing forms − h ′ r + 1 r = 8 πG Σ ( x ) (cid:20) ( α + 3 α ) ρ + 18 π ( α − α ) E (cid:21) , (35) − r = 8 πG Σ ( x ) (cid:20) (1 − α ) ρ + 18 π (4 α − E (cid:21) (36)and (cid:18) B ′ B + 12 r (cid:19) h ′ = 8 πG Σ ( x ) (cid:20) (1 − α ) ρ + 18 π (1 − α ) E (cid:21) (37)Eliminating ρ from the above equations, we ultimately getfollowing two equations − λh ′ r + 2(2 λ − r = 2 G Σ ( x ) E (38)and (cid:18) B ′ B + 12 r (cid:19) h ′ + 1 r = ( λ + 1) G (2 λ − ( x ) E (39)We see that there are two equations but three unknowns B, h and E . Similar to the assumption of interior region,let us assume that the solution of E is in the form E ( r ) = E r m . Solving equations (38) and (39), we obtain A − ( r ) ≡ h ( r ) = h + 2(2 λ − λ logr − E Gλ ( m + 1)Σ ( x ) r m +2 (40)and B ( r ) = h r Exp (cid:20)Z λ [( λ + 1) E Gr m +2 − (2 λ − ( x )] drr (2 λ − λ − ( x ) − E Gr m +2 ] (cid:21) (41)where h and h are integration constants. In this shellregion R , the range of r is satisfying D < r < D + ǫ .Since h ≪
1, so ǫ ≪
1. Under this assumption, we musthave h ≪
1. From equation (36) we obtain p = ρ = E r m πλ + (1 − λ )Σ ( x )8 πGλr (42)Also the charge density for electric field can be obtained inthe form σ ( r ) = ( m + 2) E r m − π (cid:20) h + 2(2 λ − λ logr − E Gλ ( m + 1)Σ ( x ) r m +2 (cid:21) (43)We observe that the quantities A ( r ) , B ( r ) , σ ( r ) , p, ρ de-pend on the Rastall parameter λ and Rainbow functionΣ( x ). C. Exterior Region
The exterior region R ( r > r = D + ǫ ) contains thevacuum whose EoS is given by p = ρ = 0. In this exteriorregion, using equation (25), we obtain E ( r ) = Qr (44)where Q is constant electric charge. We obtain the solu-tions as B ( r ) = kA − ( r ) = k (cid:18) − GMr + GQ (2 λ − ( x ) r (cid:19) (45)where M is the mass of the charged gravastar.So in the exterior region, the metric becomes (choose k = 1) ds = − ( x ) (cid:18) − GMr + GQ (2 λ − ( x ) r (cid:19) dt + 1Σ ( x ) (cid:18) − GMr + GQ (2 λ − ( x ) r (cid:19) − dr + r Σ ( x ) ( dθ + sin θdφ ) (46)which generates the Reissner-Nordstrom black hole inRainbow gravity [59]. For Q = 0, the above metric reducesto the Schwarzschild black hole in Rainbow gravity [60].Also we observe that the quantities A ( r ) , B ( r ) depend onthe Rastall parameter λ and Rainbow function Σ( x ). IV. PHYSICAL ASPECTS OF PARAMETERS OFCHARGED GRAVASTAR
Now we study the aspects of the physical parametersof the charged gravastar like proper length of the shell,energy and entropy within the shell. We shall also exam-ine the impact of electromagnetic field on different physi-cal features of the charged gravastar in the framework ofRastall-Rainbow gravity.
A. Proper Length of the Thin Shell
Since the radius of inner boundary of the shell of thegravastar is r = D and the radius of outer boundary of theshell is r = D + ǫ , where ǫ is the proper thickness of theshell which is assumed to be very small (i.e., ǫ ≪ ℓ = Z D + ǫD s A ( r )Σ ( x ) dr (47)Since in the shell region, the expressions of A ( r ) is compli-cated, so it is very difficult to obtain the analytical form of the above integral. So let us assume p A ( r ) = dg ( r ) dr , sofrom the above integral we can write ℓ = 1Σ( x ) Z D + ǫD dg ( r ) dr dr = 1Σ( x ) [ g ( D + ǫ ) − g ( D )] ≈ ǫ Σ( x ) dg ( r ) dr (cid:12)(cid:12)(cid:12)(cid:12) D = ǫ p A ( D )Σ( x ) (48)since ǫ ≪
1, so O ( ǫ ) ≈
0. So in the above manipulation,we have considered only the first order term of ǫ . Thus forthis approximation, the proper length will be ℓ = ǫ Σ( x ) (cid:20) h + 2(2 λ − λ logD − E Gλ ( m + 1)Σ ( x ) D m +2 (cid:21) − (49)The above result shows that the proper length of the thinshell of the gravastar is proportional to the thickness ǫ of the shell. We observe that proper length of the thinshell depends on the electric field E of the gravastar,Rastall parameter λ and Rainbow function Σ( x ). Dueto Awad et al [56] and Khodadi et al [57], here we havechosen Σ( x ) = √ x where x = E / E P l . We haveplotted the proper length ℓ vs thickness ǫ and radius D in fig. 1 and fig.2 respectively. From the figures,we have seen that the proper length ℓ increases withthe thickness ǫ of the shell of the charged gravastar butdecreases as radius D increases. On the other hand,we have also plotted the proper length ℓ vs Rastallparameter λ and test particle’s charge E in fig. 3. Fromthe figure, we have observed that the proper length ℓ veryslowly decreases as Rastall parameter λ increases andsmoothly decreases as the test particle’s charge E increases. B. Energy of the Charged Gravastar
The energy content within the shell region of the chargedgravastar is given as [12] W = Z D + ǫD πr ¯ ρ dr = 12 λG Σ ( x ) (cid:20) GE m + 3 (cid:0) ( D + ǫ ) m +3 − D m +3 (cid:1) − ǫ (3 λ − ( x ) (cid:3) (50)For the approximation ǫ ≪ W = ǫ λG Σ ( x ) (cid:2) GE D m +2 − (3 λ − ( x ) (cid:3) (51)We see that the energy content in the shell is proportionalto the thickness ( ǫ ) of the shell. Also we observe that theenergy of the gravastar depends on the electric field E ofthe gravastar, Rastall parameter λ and Rainbow functionΣ( x ). We have plotted the energy content W in the shellvs thickness ǫ and radius D in fig. 4 and fig.5 respectively.From the figures, we have seen that the the energy content W in the shell increases with the thickness ǫ of the shellof the charged gravastar as well as the radius D . On theother hand, we have also plotted the energy content W inthe shell vs Rastall parameter λ and test particle’s charge E in fig. 6. From the figure, we have observed that theenergy content W in the shell decreases as increase of theRastall parameter λ and the test particle’s charge E . C. Entropy of the Charged Gravastar
Entropy is the disorderness within the body of a gravas-tar. Mazur and Mottola [1, 2] have shown that the entropydensity in the interior region R of the gravastar is zero.But, the entropy within the thin shell can be described by[1] S = Z D + ǫD πr s ( r ) s A ( r )Σ ( x ) dr (52)where s ( r ) is the entropy density corresponding to the spe-cific temperature T ( r ) and which can be defined as s ( r ) = γ k B T ( r )4 π ¯ h = γk B ¯ h r p ( r )2 π (53)where γ is dimensionless constant. So entropy within thethin shell can be written as S = γk B ¯ h Σ( x ) Z D + ǫD r p πp ( r ) A ( r ) dr = γk B ¯ h √ λG Σ( x ) Z D + ǫD r (cid:2) GE r m +2 + (1 − λ )Σ ( x ) (cid:3) × (cid:20) h + 2(2 λ − λ logr − E Gλ ( m + 1)Σ ( x ) r m +2 (cid:21) − dr (54)Now it is very difficult to obtain analytical form of theabove integral. So using the approximation ǫ ≪
1, we canobtain
S ≈ ǫγk B ¯ h Σ( x ) D p πp ( D ) A ( D ) ≈ ǫγk B D ¯ h √ λG Σ( x ) (cid:2) GE D m +2 + (1 − λ )Σ ( x ) (cid:3) × (cid:20) h + 2(2 λ − λ logD − E Gλ ( m + 1)Σ ( x ) D m +2 (cid:21) − (55)This result shows that the entropy in the shell of thecharged gravastar is proportional to the thickness ǫ ofthe shell. We observe that the entropy depends on theelectric field E of the gravastar, Rastall parameter λ andRainbow function Σ( x ). We have plotted the entropy S within the shell vs thickness ǫ and radius D in fig. 7 andfig.8 respectively. From the figures, we have seen that thethe entropy S within the shell increases with the thickness ǫ of the shell of the charged gravastar as well as the radius Ε { Fig. 1 represents the plot proper length ℓ vs thickness ǫ . { Fig. 2 represents the plot of proper length ℓ vs radius D . { Λ E Fig. 3 represents the plot of proper length ℓ vs Rastallparameter λ and test particle’s charge E . D . On the other hand, we have also plotted the entropy S within the shell vs Rastall parameter λ and test particle’scharge E in fig. 9. From the figure, we have observed thatthe entropy S within the shell decreases as increase of theRastall parameter λ and the test particle’s charge E . V. JUNCTION CONDITIONS BETWEENINTERIOR AND EXTERIOR REGIONS
Since gravastar consists of three regions: interior region,thin shell region and exterior region, so according to theDarmois-Israel formalism [61–63] we want to study thematching between the surfaces of the interior and exte-rior regions. We denote P is the junction surface which islocated at r = D . In the Rainbow gravity, we consider the Ε W Fig. 4 represents the plot of energy content W vs thickness ǫ . W Fig. 5 represents the plot of energy content W vs radius D . W Λ E Fig. 6 represents the plot of energy content W vs Rastallparameter λ and test particle’s charge E . metric on the junction surface as in the form ds = − f ( r )Π ( x ) dt + 1Σ ( x ) f ( r ) dr + r Σ ( x ) ( dθ +sin θdφ )(56)where the metric coefficients are continuous at the junctionsurface P , but their derivatives might not be continuousat P . In the joining surface S , the surface tension andsurface stress energy may be resolved by the discontinuityof the extrinsic curvature of S at r = D . The expressionof the stress-energy surface S ij is defined by the Lanczosequation [64] (with the help of Darmois-Israel formalism) S ij = − π ( η ij − δ ij η kk ) (57)where η ij = K + ij − K − ij . Here K ij denotes the extrinsiccurvature. Here the signs “ − ” and “ + ” respectivelycorrespond to the interior and the exterior regions of thegravastar. So η ij describes the discontinuous surfaces inthe second fundamental forms of the extrinsic curvatures. Ε S Fig. 7 represents the plot of entropy S vs thickness ǫ . S Fig. 8 represents the plot of entropy S vs radius D . S Λ E Fig. 9 represents the plot of entropy S vs Rastall parameter λ and test particle’s charge E . The extrinsic curvatures on the both surfaces of the shellregion can be described by K ± ij = (cid:20) − n ± ν (cid:26) ∂ x ν ∂ξ i ∂ξ j + Γ ναβ ∂x α ∂ξ i ∂x β ∂ξ j (cid:27)(cid:21)P (58)where ξ i represent the intrinsic coordinates on the shell, n ± ν describe the unit normals to the surface P of the gravastar,defined by n ν n ν = −
1. For the above metric, we can obtain n ± ν = ± (cid:20) g αβ ∂f∂x α ∂f∂x β (cid:21) − ∂f∂x ν (59)According to the Lanczos equation, the stress-energy sur-face tensor can be written as S ij = diag ( − ̺, P , P , P ) where ̺ is the surface energy density and P is the surface pressure.Using the matching conditions, in the charged gravastarmodel, the surface energy density and the surface pressurecan be obtained as ̺ ( D ) = 14 πD (cid:0) − q ( x ) D + q ( x ) D m +2 (cid:1) − πD (cid:18) − GMD + q ( x ) D (cid:19) (60)and P ( D ) = 18 πD (cid:18) − GMD + q ( x ) D (cid:19) − πD (cid:0) − q ( x ) D + q ( x ) D m +2 (cid:1) + 116 π (cid:18) GMD − q ( x ) D (cid:19) (cid:18) − GMD + q ( x ) D (cid:19) − − π (cid:0) − q ( x ) D + ( m + 1) q ( x ) D m +1 (cid:1) × (cid:0) − q ( x ) D + q ( x ) D m +2 (cid:1) − (61)where q ( x ) = GQ (2 λ − ( x ) , q ( x ) = 8 πGk λ − ( x ) ,q ( x ) = 2 Gm (2 m + 3)(2 λ − ( x ) . (62)We have drawn the plots of surface energy density ̺ ( D )and surface pressure P ( D ) vs radius D in fig. 10 and fig.12. From these figures we have seen that the surface energydensity ̺ ( D ) increases and surface pressure P ( D ) decreasesas radius D increases. On the other hand, we have drawnthe plots of surface energy density ̺ ( D ) and surface pres-sure P ( D ) vs Rastall parameter λ and test particle’s charge E in fig. 11 and fig. 13. We have observed that if Rastallparameter λ and test particle’s charge E increase then sur-face energy density ̺ ( D ) decreases but surface pressure P ( D ) increases. Also we have seen that surface pressure P ( D ) is always keep negative sign. A. Equation of State
The equation of state parameter w ( D ) (at r = D ) canbe described as w ( D ) = P ( D ) ̺ ( D ) (63)So the equation of state parameter can be obtained in thefollowing form w ( D ) = −
12 + 14 (cid:20)(cid:18)
GMD − q ( x ) D (cid:19) × (cid:18) − GMD + q ( x ) D (cid:19) − − (cid:0) − q ( x ) D + ( m + 1) q ( x ) D m +1 (cid:1) × (cid:0) − q ( x ) D + q ( x ) D m +2 (cid:1) − i × h(cid:0) − q ( x ) D + q ( x ) D m +2 (cid:1) − (cid:18) − GMD + q ( x ) D (cid:19) − (64) We have plotted the equation of state parameter w ( D ) vsradius D in fig. 14 and seen that w ( D ) increases as radius D increases. Also we have plotted the equation of stateparameter w ( D ) vs Rastall parameter λ and test particle’scharge E in fig. 15 and it decreases as Rastall parameter λ and test particle’s charge E increase but w ( D ) is alwayskeep negative sign. B. Mass
The mass M of the thin shell of the charged gravastarcan be obtained from the following formula M = 4 πD ̺ ( D ) (65)which can be expressed as in the form M = D h(cid:0) − q ( x ) D + q ( x ) D m +2 (cid:1) − (cid:18) − GMD + q ( x ) D (cid:19) (66)So the total mass M of the charged gravastar in terms ofthe mass of the thin shell can be expressed as M = D G + q ( x )2 GD − D G (cid:20)(cid:0) − q ( x ) D + q ( x ) D m +2 (cid:1) − M D (cid:21) (67)We see that the total mass M of the gravastar will be lessthan D + q ( x )2 GD . We have plotted the mass M of the thinshell of the charged gravastar vs radius D in fig. 16 andseen that the mass M increases as radius D increases. Alsowe have plotted the the mass M vs Rastall parameter λ and test particle’s charge E in fig. 17 and it decreases asRastall parameter λ and test particle’s charge E increase. C. Stability
Poisson et al [65] have examined the linearized stabilityfor thin shell wormhole. Lobo et al [66] have examined thelinearized stability for thin shell wormhole with cosmologi-cal constant. Ovgun et al [28] have examined the stabilityof charged thin-shell gravastar. Also Yousaf et al [43] havediscussed the stability of the gravastar in f ( R, T ) gravity.Motivated by their work, we want to analyze the stabilityof the charged gravastar in Rastall-Rainbow gravity. Forthis purpose, define a parameter η as follows [65]: η ( D ) = P ′ ( D ) ̺ ′ ( D ) (68)The parameter η is interpreted as the squared speed ofsound satisfying 0 ≤ η ≤ η vs radius D in fig. 18and seen that η decreases as radius D increases. Also we Ρ H D L Fig. 10 represents the plot of surface energy density ̺ ( D ) vsradius D . Ρ H D L Λ E Fig. 11 represents the plot of surface energy density ̺ ( D ) vsRastall parameter λ and test particle’s charge E . - - - - - - -
500 D P H D L Fig. 12 represents the plot of surface pressure P ( D ) vs radius D . P H D L Λ E - - - Fig. 13 represents the plot of surface pressure P ( D ) vs Rastallparameter λ and test particle’s charge E . - - - - - - - - w H D L Fig. 14 represents the plot of equation of state parameter w ( D ) vs radius D . w H D L Λ E - - - - - Fig. 15 represents the plot of equation of state parameter w ( D ) vs Rastall parameter λ and test particle’s charge E . M Fig. 16 represents the plot of the mass of the thin shell M vsradius D . M Λ E Fig. 17 represents the plot of the mass of the thin shell M vsRastall parameter λ and test particle’s charge E . Η H D L Fig. 18 represents the plot of η ( D ) vs radius D . Η H D L Λ E Fig. 19 represents the plot of η ( D ) vs Rastall parameter λ andtest particle’s charge E . have plotted η vs Rastall parameter λ and test particle’scharge E in fig. 19 and it decreases as Rastall parameter λ and test particle’s charge E increase. Figures show that η always keep positive sign which allows the stability of thegravastar model. VI. DISCUSSIONS
In this work, we have considered the spherically sym-metric stellar system in the contexts of Rastall-Rainbowgravity theory in presence of isotropic fluid source withelectro-magnetic field. The Einstein-Maxwell’s fieldequations have been written in the framework of Rastall-Rainbow gravity. Next we have discussed the geometry ofcharged gravastar model. The gravastar consists of threeregions: interior region, thin shell region and exteriorregion. In the interior region, the gravastar follows theequation of sate (EoS) p = − ρ and we have found thesolutions of all physical quantities like energy density,pressure, electric field, charge density, gravitational massand metric coefficients. In the exterior region, we haveobtained the exterior Riessner-Nordstrom solution for vacuum model ( p = ρ = 0). In the shell region, the fluidsource follows the EoS p = ρ (ultra-stiff perfect fluid).In this region, since we have assumed that the interiorand exterior regions join together at a place, so theintermediate region must be thin shell and the thicknessof the shell of the gravastar is infinitesimal. So the thinshell follows with the limit h ( ≡ A − ) ≪
1. Under thisapproximation, we have found the analytical solutionswithin the thin shell of the gravastar. The physicalquantities like the proper length of the thin shell, energycontent and entropy inside the thin shell of the chargedgravastar have been computed and we have shown thatthey are directly proportional to the proper thickness ofthe shell ( ǫ ) due to the approximation ( ǫ ≪ λ and Rainbow function Σ( x ) (which dependson energy of the test particle). From figures 1 - 9, wehave seen that the proper length of the thin shell, energycontent and entropy inside the thin shell of the chargedgravastar increase as thickness increases. Also the properlength decreases as radius increases. The energy contentand entropy inside the thin shell always increase as radiusincreases. Moreover, if Rastall parameter λ and test parti-cle’s charge E both increase then the proper length of thethin shell, energy content and entropy inside the thin shellall decrease. According to the Darmois-Israel formalism,we have studied the matching between the surfaces ofinterior and exterior regions of the charged gravastar andusing the matching conditions, the surface energy densityand the surface pressure have been obtained. Also theequation of state parameter on the surface, mass of thethin shell have been obtained and the total mass of thecharged gravastar have been expressed in terms of thethin shell mass. From figures 10, 14, 16 we have observedthat the surface density, equation of state and mass of thethin shell increase as radius increases but from figure 12,we have seen that the surface pressure always decreasesas radius increases but keeps negative sign and hence theequation of state keeps negative sign. Also the figures 11,15, 17 show that the surface density, equation of stateand mass of the thin shell decrease as the increase ofboth Rastall parameter λ and test particle’s charge E .But figure 13 shows the surface pressure increases as theincrease of both Rastall parameter λ and test particle’scharge E . Finally, we have explored the stable regions ofthe charged gravastar in Rastall-Rainbow gravity in figures18 and 19. The figures show that η always decreases butkeeps in positive sign as the increase of radius, Rastallparameter λ and test particle’s charge E . [1] P. Mazur and E. Mottola, Report number: LA-UR-01- 5067(arXiv:gr-qc/0109035).[2] P. Mazur and E. Mottola, Proc. Natl. Acad. Sci. (USA)101, 9545 (2004).[3] M. Visser and D. L. Wiltshire, Class. Quant. Grav. 21, 1135 (2004).[4] A. DeBenedictis, D. Horvat, S. Ilijic, S. Kloster and K. S.Viswanathan, Class. Quant. Grav. 23, 2303 (2006).[5] C. Cattoen, T. Faber and M. Visser, Class. Quant. Grav.22, 4189 (2005).1