Charged Gravastars in Rastall-Rainbow Gravity

Home , Rainbow gravity theory

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Rastall gravity theory is proposed by Rastall [45], which above equation can be written as is one of the alternative of modified gravity theory by ν ¯ ν modification the Einstein’s general relativity. Neutron Tµ λδµR = 0 (2) − ;ν Stars in Rastall Gravity have been obtained by Oliveira et al [46]. A model of quintessence compact stars in the So for Rastall’s gravity, the Einstein’s equation can be Rastall’s theory of gravity has been obtained by Abhas modified to the form et al [47]. Isotropic compact star model in Rastall theory ν 1 ν ν ¯ ν admitting conformal motion has also been obtained in Rµ δµR =8πG Tµ λδµR (3) [48]. Anisotropic compact star model in the Rastall theory − 2 − of gravity has been discussed in [49]. Gravity’s rainbow which can be simplified to the form [50] is a distortion of space-time which is an extension λ of the doubly special relativity for curved space-times. Rν δν R =8πGT ν (4) The properties of neutron stars and dynamical stability µ − 2 µ µ conditions in the modified TOV in gravity’s rainbow Now the trace of the energy-momentum tensor is given by have been investigated by Hendi et al [51]. The gravity’s rainbow and compact star models have been studied in (1 2λ)R [52]. The rainbow’s star models have also been studied in T = − (5) 8πG [53]. Recently, Mota et al [54] have studied the neutron star model in the framework of Rastall-Rainbow theories So the above Einstein’s equation in Rastall’s gravity can of gravity. be written as 1 (1 λ) The main motivation of the work is to study the gravas- Rν δν R =8πG T ν − δν T (6) µ − 2 µ µ − 2(1 2λ) µ tar system in the framework of Rastall-Rainbow gravity − with the isotropic fluid and electromagnetic source and Now assume that the fluid source is composed of normal examine the nature of physical parameters and stability of matter and electro-magnetic field. So the energy momen- the gravastar. The organization of the work is as follows: tum tensor can be written as In section II, we present the Rastall-Rainbow gravity theory with electromagnetic field. Here we write the T = T M + T EM (7) Einstein-Maxwell field equations for spherically symmetric µν µν µν steller metric in the framework of Rastall-Rainbow gravity. where the energy-momentum tensor for normal matter is Section III deals with the geometry of gravastar and we given by compute the solutions in the three regions of gravastar M model. In section IV, we analyze the physical aspects of Tµν = (ρ + p)uµuν + pgµν (8) the parameters of gravastar model. In section V, we inves- µ ν tigate the matching between interior and exterior regions. where u is the fluid four-velocity satisfying uµu = 1, ρ Due to the junction conditions, we compute the equation and p are the energy density and pressure of fluid. Further,− of state, mass of the thin shell region and examine the the energy momentum tensor for electromagnetic field is stability of the gravastar. Finally some physical analysis given by [55] and fruitful conclusions of the work are drawn in section VI. EM 1 δω 1 δω T = (g FµδFων gµν FδωF ) (9) µν −4π − 4

where Fµν is the Maxwell ﬁeld tensor deﬁned as in the II. RASTALL-RAINBOW GRAVITY form:

In Einstein’s General Relativity (GR), the conserva- Fµν =Φν,µ Φµ,ν (10) ν − tion law of energy-momentum tensor is Tµ;ν = 0. The Rastall’s Gravity is a generalization of General Relativity, and Φµ is the four potential. The corresponding equations where Rastall [45] proposed the modiﬁcation of conserva- for Maxwell’s electromagnetic ﬁeld are given by tion law of energy-momentum tensor in curved space-time µν µ (√ g F ),ν =4πJ √ g , F = 0 (11) and which is given by [54] − − [µν,δ] where J µ is the current four-vector satisfying J µ = σuµ, ν ¯ Tµ;ν = λR,µ (1) the parameter σ is the charge density.

¯ 1 λ where λ = 16−πG with λ is a constant called Rastall param- Magueijo and Smolin [50] have proposed gravity’s Rain- eter, which measures the deviation from GR and describes bow, which is an extension of the doubly special relativity the affinity of the matter field to couple with geometry. For for curved space-times. Gravity’s rainbow is a distortion λ = 1, the usual conservation law can be restored. Also for of space-time induced by two arbitrary functions Π(x) and flat space-time, the Ricci scalar R = 0 and we may also Σ(x) (called the Rainbow functions) satisfying the usual conservation law. So for the effect of Rastall’s gravity, λ = 1 and the space-time must be non-flat. The 2Π2(x) υ2Σ2(x)= m2 (12) 6 E − 3

5 where x = / Pl. Here , υ, m and Pl = √¯hc /G are We observe that α1 + α2 = 1 and λ = 1/2. Adding equa- the energy, momentum,E E massE of a test particleE and Planck tions (15) and (16), we obtain 6 energy respectively. Awad et al [56] and Khodadi et al [57] 1 A B 8πG have chosen Π(x) = 1 and Σ(x) = √1+ x2 to study the ′ + ′ =8πG(¯ρ +¯p )= (ρ + p) (22) rA A B 1 Σ2(x) solutions corresponding to a nonsingular universe. Also to study of gamma ray burst, the exponential form of rainbow From equation (15), we obtain [54] has been applied in [56, 58]. In the absence of the test particles, the Rainbow functions are satisfying 1 2GM(r) − A(r)= 1 (23) lim Π(x)=1, lim Σ(x)=1 (13) − r x 0 x 0 → → where the gravitational mass is Mota et al [54] have merged the Rastall’s gravity with r Rainbow’s gravity and applied the Rastall-Rainbow gravity M(r)= 4πr2ρ¯(r)dr (24) theory in the neutron star formation. We consider the 0 spherically symmetric metric describing the interior space- Z time of a star in Rainbow gravity as [54] From the modification of conservation law of energy- momentum tensor, we obtain 2 2 B(r) 2 A(r) 2 r 2 2 2 ds = dt + dr + (dθ + sin θdφ ) B′ 1 4 2 −Π2(x) Σ2(x) Σ2(x) p′ + (ρ + p) = (r E )′ (25) 2B 8πr4 (14) where A(r) and B(r) are functions of r. Since Π(x) and the electric field E is as follows and Σ(x) depend on x = / Pl and is independent of r r, so Π(x) and Σ(x) are independentE E E of r. The metric 1 2 A(r) E(r)= 2 4πr σ(r) dr (26) coefficients depend of the energy of the test particle. So r 0 Σ(x) Z p the geometry of the space-time becomes energy dependent. σ√A It should be noted that the metric coordinates do not The term Σ(x) inside the above integral is known as the depend on the energy of the particle. volume charge density.

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 be III. GRAVASTAR written in the form [54] Here, we will derive the solutions of the field equa- A 1 1 ′ + =8πGρ¯ , (15) tions for charged gravastar (charge is generated by electro- rA2 − r2A r2 magnetic field) and analyze its physical as well as geometrical interpretations. Since there are three regions of B 1 1 the gravastar, so the geometrical regions of the gravastar ′ + =8πGp¯ , (16) having a finite extremely thin width within the regions rAB r2A − r2 1 D = r1

1 1 The interior region R1 (0 r < r1 = D) of the gravastar p¯ = α ρ + (1 3α )p + (4α 1)E2 , (19) ≤ 1 Σ2(x) 2 − 2 8π 2 − follows the EoS p = ρ. From equation (17), we obtain the relation − and 1 B(r)= kA− (r) (27) 1 1 p¯ = α ρ + (1 3α )p + (1 2α )E2 (20) 2 Σ2(x) 2 − 2 8π − 2 where k is constant > 0. There are 4 equations and 5 unknown functions A, B, ρ, p, E in the system. So one with function is free to us. Here we may consider E is free function of r and is chosen by 3λ 1 λ 1 α1 = − , α2 = − (21) m 2(2λ 1) 2(2λ 1) E(r)= E0r (28) − − 4 where m is positive constant and E0 is function of x. obtain the analytical solution within the thin shell. Under this approximation (we can set h 0 to the leading order) Using equation (25), we obtain with p = ρ, the ﬁeld equations (15)≈ - (17) reduce to the following forms p = ρ = k r2m k (29) − 1 − 2 2 h′ 1 8πG 1 2 (m+2)E 0 + 2 = 2 (α1 +3α2)ρ + (α1 3α2)E , where k1 = 8πm and k2 is positive constant. From − r r Σ (x) 8π − (24), we obtain (35)

2 1 8πk2 3 2E0 2m+3 1 8πG 1 M(r)= r r = (1 2α )ρ + (4α 1)E2 (36) 2(2λ 1)Σ2(x) 3 − m(2m + 3) 2 2 2 2 −r Σ (x) − 8π − − (30) From equation (23), we obtain the solution and

1 8πGk2 2 B′ 1 ′ 8πG 1 B(r)= kA− (r)= k 1 r + h = (1 2α )ρ + (1 2α )E2 − 3(2λ 1)Σ2(x) 4B 2r Σ2(x) − 2 8π − 2 − 2GE2 (37) + 0 r2m+2 (31) m(2m + 3)(2λ 1)Σ2(x) Eliminating ρ from the above equations, we ultimately get − following two equations So the metric becomes (choose k = 1) λh′ 2(2λ 1) 2G + − = E2 (38) 1 8πGk2 − r r2 Σ2(x) ds2 = 1 r2 −Π2(x) − 3(2λ 1)Σ2(x) 2 − and 2GE0 2m+2 2 + 2 r dt m(2m + 3)(2λ 1)Σ (x) B 1 ′ 1 (λ + 1)G − ′ + h + = E2 (39) 1 8πGk2 4B 2r r2 (2λ 1)Σ2(x) + 1 r2 − Σ2(x) − 3(2λ 1)Σ2(x) − 1 We see that there are two equations but three unknowns 2GE2 − + 0 r2m+2 dr2 B, h and E. Similar to the assumption of interior region, m(2m + 3)(2λ 1)Σ2(x) let us assume that the solution of E is in the form E(r)= 2 − m r E0r . Solving equations (38) and (39), we obtain + (dθ2 + sin2 θdφ2) (32) Σ2(x) 2 1 2(2λ 1) E0 G 2m+2 A− (r) h(r)= h + − logr r The charge density for electric ﬁeld is obtained in the form: ≡ 1 λ −λ(m + 1)Σ2(x) (40) (m + 2)E rm 1 8πGk σ(r)= 0 − 1 2 r2 and 4π − 3(2λ 1)Σ2(x) − 2 2m+2 2 1 h2 4λ[(λ + 1)E Gr (2λ 1)Σ (x)] dr 2GE2 2 B(r)= Exp 0 − − + 0 r2m+2 (33) r2 r(2λ 1)[2(2λ 1)Σ2(x) 2E2Gr2m+2] m(2m + 3)(2λ 1)Σ2(x) Z − − − 0 − (41) where h and h are integration constants. In this shell Also the gravitational mass of the interior region of the 1 2 region R , the range of r is satisfying D

dg(r) C. Exterior Region of the above integral. So let us assume A(r) = dr , so from the above integral we can write p The exterior region R (r > r = D + ǫ) contains the 3 2 1 D+ǫ dg(r) 1 vacuum whose EoS is given by p = ρ = 0. In this exterior ℓ = dr = [g(D + ǫ) g(D)] Σ(x) dr Σ(x) − region, using equation (25), we obtain ZD ǫ dg(r) ǫ A(D) Q = (48) E(r)= (44) ≈ Σ(x) dr Σ(x) r2 D p

since ǫ 1, so O(ǫ 2) 0. So in the above manipulation, where Q is constant electric charge. We obtain the solu- ≪ ≈ tions as we have considered only the first order term of ǫ. Thus for this approximation, the proper length will be 2 1 2GM GQ B(r)= kA− (r)= k 1 + 1 − r (2λ 1)Σ2(x)r2 ǫ 2(2λ 1) E2G − 2 − ℓ = h + − logD 0 D2m+2 (45) Σ(x) 1 λ − λ(m + 1)Σ2(x) where M is the mass of the charged gravastar. (49) The above result shows that the proper length of the thin So in the exterior region, the metric becomes (choose shell of the gravastar is proportional to the thickness ǫ k = 1) of the shell. We observe that proper length of the thin shell depends on the electric field E of the gravastar, 1 2GM GQ2 0 ds2 = 1 + dt2 Rastall parameter λ and Rainbow function Σ(x). Due −Π2(x) − r (2λ 1)Σ2(x)r2 − to Awad et al [56] and Khodadi et al [57], here we have 1 2 1 2GM GQ2 − chosen Σ(x) = √1+ x where x = / Pl. We have + 1 + dr2 E E Σ2(x) − r (2λ 1)Σ2(x)r2 plotted the proper length ℓ vs thickness ǫ and radius − D in fig. 1 and fig.2 respectively. From the figures, r2 + (dθ2 + sin2 θdφ2) (46) we have seen that the proper length ℓ increases with Σ2(x) the thickness ǫ of the shell of the charged gravastar but decreases as radius D increases. On the other hand, which generates the Reissner-Nordstrom black hole in we have also plotted the proper length ℓ vs Rastall Rainbow gravity [59]. For Q = 0, the above metric reduces parameter λ and test particle’s charge in fig. 3. From to the Schwarzschild black hole in Rainbow gravity [60]. the figure, we have observed that the properE length ℓ very Also we observe that the quantities A(r), B(r) depend on slowly decreases as Rastall parameter λ increases and the Rastall parameter λ and Rainbow function Σ(x). smoothly decreases as the test particle’s charge increases. E

IV. PHYSICAL ASPECTS OF PARAMETERS OF CHARGED GRAVASTAR B. Energy of the Charged Gravastar

Now we study the aspects of the physical parameters The energy content within the shell region of the charged of the charged gravastar like proper length of the shell, gravastar is given as [12] energy and entropy within the shell. We shall also examine the impact of electromagnetic field on different physi- D+ǫ cal features of the charged gravastar in the framework of = 4πr2ρ¯ dr W Rastall-Rainbow gravity. ZD 1 2GE2 = 0 (D + ǫ)2m+3 D2m+3 2λGΣ2(x) 2m +3 − A. Proper Length of the Thin Shell ǫ(3λ 2)Σ2(x) (50) − − Since the radius of inner boundary of the shell of the For the approximation ǫ 1 due to thin shell region of the gravastar is r = D and the radius of outer boundary of the gravastar, we may obtain≪ shell is r = D + ǫ, where ǫ is the proper thickness of the ǫ shell which is assumed to be very small (i.e., ǫ 1). So 2 2m+2 2 = 2 2GE0 D (3λ 2)Σ (x) (51) the stiff perfect fluid propagates between two boundaries≪ W 2λGΣ (x) − − of the thin shell region of the gravastar. Now, the proper We see that the energy content in the shell is proportional length of the shell is described by [1] to the thickness (ǫ) of the shell. Also we observe that the energy of the gravastar depends on the electric field E0 of D+ǫ A(r) the gravastar, Rastall parameter λ and Rainbow function ℓ = 2 dr (47) D sΣ (x) Σ(x). We have plotted the energy content in the shell Z vs thickness ǫ and radius D in fig. 4 and fig.5W respectively. Since in the shell region, the expressions of A(r) is compli- From the figures, we have seen that the the energy content cated, so it is very difficult to obtain the analytical form in the shell increases with the thickness ǫ of the shell W 6 of the charged gravastar as well as the radius D. On the other hand, we have also plotted the energy content in 0.04 W the shell vs Rastall parameter λ and test particle’s charge 0.03 in fig. 6. From the figure, we have observed that the E { energy content in the shell decreases as increase of the 0.02 Rastall parameterW λ and the test particle’s charge . E 0.01

0.00 0.00 0.02 0.04 0.06 0.08 0.10 Ε C. Entropy of the Charged Gravastar Fig. 1 represents the plot proper length ℓ vs thickness ǫ. Entropy is the disorderness within the body of a gravastar. Mazur and Mottola [1, 2] have shown that the entropy 0.032 density in the interior region R1 of the gravastar is zero. 0.030 But, the entropy within the thin shell can be described by 0.028 [1] { 0.026 D+ǫ 2 A(r) 0.024 = 4πr s(r) 2 dr (52) S D s Σ (x) 0.022 Z 0 1 2 3 4 where s(r) is the entropy density corresponding to the spe- D ciﬁc temperature T (r) and which can be deﬁned as Fig. 2 represents the plot of proper length ℓ vs radius D. { γ2k2 T (r) γk p(r) s(r)= B = B (53) 4π¯h ¯h r 2π where γ is dimensionless constant. So entropy within the thin shell can be written as 0.06

D+ǫ 0.05 10 γkB 2 0.04 = r 8πp(r) A(r) dr 0.03 8 S ¯hΣ(x) D 0.02 Z 6 D+pǫ 1 1 E γkB 2 2m+2 2 2 = r GE0 r + (1 2λ)Σ (x) 2 4 ¯h√λG Σ(x) D − × 3 Z Λ 1 4 2 2(2λ 1) E2G − 2 h + − logr 0 r2m+2 dr 5 1 λ − λ(m + 1)Σ2(x) Fig. 3 represents the plot of proper length ℓ vs Rastall (54) parameter λ and test particle’s charge E. Now it is very difficult to obtain analytical form of the above integral. So using the approximation ǫ 1, we can ≪ obtain D. On the other hand, we have also plotted the entropy S ǫγk within the shell vs Rastall parameter λ and test particle’s B D2 8πp(D) A(D) charge in fig. 9. From the figure, we have observed that S ≈ ¯hΣ(x) the entropyE within the shell decreases as increase of the p 1 S ǫγkBD 2 2m+2 2 2 Rastall parameter λ and the test particle’s charge . GE0 D + (1 2λ)Σ (x) E ≈ ¯h√λG Σ(x) − × 1 2(2λ 1) E2G − 2 h + − logD 0 D2m+2 1 λ − λ(m + 1)Σ2(x) (55) V. JUNCTION CONDITIONS BETWEEN This result shows that the entropy in the shell of the INTERIOR AND EXTERIOR REGIONS charged gravastar is proportional to the thickness ǫ of the shell. We observe that the entropy depends on the electric field E0 of the gravastar, Rastall parameter λ and Since gravastar consists of three regions: interior region, Rainbow function Σ(x). We have plotted the entropy thin shell region and exterior region, so according to the within the shell vs thickness ǫ and radius D in fig. 7 andS Darmois-Israel formalism [61–63] we want to study the fig.8 respectively. From the figures, we have seen that the matching between the surfaces of the interior and exte- the entropy within the shell increases with the thickness rior regions. We denote is the junction surface which is ǫ of the shellS of the charged gravastar as well as the radius located at r = D. In the Rainbow gravity, we consider the P 7

0.5 2.5

2.0 0.4

1.5 0.3 S W 1.0 0.2

0.5 0.1

0.0 0.0 0.00 0.02 0.04 0.06 0.08 0.10 0.00 0.02 0.04 0.06 0.08 0.10 Ε Ε Fig. 4 represents the plot of energy content W vs thickness ǫ. Fig. 7 represents the plot of entropy S vs thickness ǫ.

1200

0.5 1000

0.4 800

0.3

600 S W

400 0.2

200 0.1

0 0.0 0 1 2 3 4 0 1 2 3 4 5 D D

Fig. 5 represents the plot of energy content W vs radius D. Fig. 8 represents the plot of entropy S vs radius D. W S

3 0.15 10 10 2 0.10

1 8 0.05 8

0 0.00 6 6 1 E 1 E 2 4 2 4 3 3 Λ Λ 4 2 4 2

5 5

Fig. 6 represents the plot of energy content W vs Rastall Fig. 9 represents the plot of entropy S vs Rastall parameter λ parameter λ and test particle’s charge E. and test particle’s charge E. metric on the junction surface as in the form The extrinsic curvatures on the both surfaces of the shell region can be described by 2 2 f(r) 2 1 2 r 2 2 2 2 α β ds = dt + dr + (dθ +sin θdφ ) ∂ xν ν ∂x ∂x 2 2 2 K± = n± +Γ (58) −Π (x) Σ (x)f(r) Σ (x) ij − ν ∂ξi∂ξj αβ ∂ξi ∂ξj (56) where the metric coefficients are continuous at the junction i where ξ represent the intrinsic coordinates on theP shell, nν± surface , but their derivatives might not be continuous describe the unit normals to the surface of the gravastar, at . In the joining surface S, the surface tension and ν defined by nν n = 1. For the above metric, we can obtain P − surface stress energy may be resolved by the discontinuity 1P P 2 of the extrinsic curvature of S at r = D. The expression αβ ∂f ∂f − ∂f n± = g (59) of the stress-energy surface Sij is defined by the Lanczos ν ± ∂xα ∂xβ ∂xν equation [64] (with the help of Darmois-Israel formalism) According to the Lanczos equation, the stress-energy sur- 1 face tensor can be written as Si = diag( ̺, , , ) where Si = (ηi δi ηk) (57) j − P P P j −8π j − j k ̺ is the surface energy density and is the surface pressure. Using the matching conditions, inP the charged gravastar + where ηij = Kij Kij−. Here Kij denotes the extrinsic model, the surface energy density and the surface pressure curvature. Here the− signs “ ” and “ + ” respectively can be obtained as correspond to the interior and− the exterior regions of the gravastar. So ηij describes the discontinuous surfaces in 1 1 ̺(D)= 1 q (x)D2 + q (x)D2m+2 2 the second fundamental forms of the extrinsic curvatures. 4πD − 2 3 8

1 1 2GM q (x) 2 We have plotted the equation of state parameter w(D) vs 1 + 1 (60) −4πD − D D2 radius D in ﬁg. 14 and seen that w(D) increases as radius D increases. Also we have plotted the equation of state and parameter w(D) vs Rastall parameter λ and test particle’s

1 charge in fig. 15 and it decreases as Rastall parameter 1 2GM q (x) 2 E (D)= 1 + 1 λ and test particle’s charge increase but w(D) is always P 8πD − D D2 keep negative sign. E 1 1 1 q (x)D2 + q (x)D2m+2 2 −8πD − 2 3 1 B. Mass 1 GM q (x) 2GM q (x) − 2 + 1 1 + 1 16π D2 − D3 − D D2 The mass of the thin shell of the charged gravastar 1 M q (x)D + (m + 1)q (x)D2m+1 can be obtained from the following formula −16π − 2 3 × 1 2 2 2m+2 2 =4πD ̺(D) (65) 1 q2(x)D + q3(x)D − (61) M − where which can be expressed as in the form 1 2 2 2m+2 2 GQ 8πGk2 = D 1 q2(x)D + q3(x)D q1(x)= 2 , q2(x)= 2 , M − (2λ 1)Σ (x) 3(2λ 1)Σ (x) 1 − − h 2 2G 2GM q1(x) q3(x)= . (62) 1 + 2 (66) m(2m + 3)(2λ 1)Σ2(x) − − D D # − We have drawn the plots of surface energy density ̺(D) So the total mass M of the charged gravastar in terms of and surface pressure (D) vs radius D in fig. 10 and fig. the mass of the thin shell can be expressed as 12. From these figuresP we have seen that the surface energy density ̺(D) increases and surface pressure (D) decreases D q1(x) P M = + as radius D increases. On the other hand, we have drawn 2G 2GD the plots of surface energy density ̺(D) and surface pres- 1 2 D 2 2m+2 2 sure (D) vs Rastall parameter λ and test particle’s charge 1 q2(x)D + q3(x)D M (67) −2G − − D inP fig. 11 and fig. 13. We have observed that if Rastall E parameter λ and test particle’s charge increase then sur- We see that the total mass M of the gravastar will be less E 2 face energy density ̺(D) decreases but surface pressure D q1 x than + ( ) . We have plotted the mass of the thin (D) increases. Also we have seen that surface pressure 2GD shell of the charged gravastar vs radius D Min fig. 16 and P(D) is always keep negative sign. P seen that the mass increases as radius D increases. Also we have plotted theM the mass vs Rastall parameter λ M A. Equation of State and test particle’s charge in fig. 17 and it decreases as Rastall parameter λ and testE particle’s charge increase. E The equation of state parameter w(D) (at r = D) can be described as C. Stability (D) w(D)= P (63) ̺(D) Poisson et al [65] have examined the linearized stability for thin shell wormhole. Lobo et al [66] have examined the So the equation of state parameter can be obtained in the linearized stability for thin shell wormhole with cosmologi- following form cal constant. Ovgun et al [28] have examined the stability of charged thin-shell gravastar. Also Yousaf et al [43] have 1 1 GM q (x) 1 discussed the stability of the gravastar in f(R, ) gravity. w(D)= + 2 3 −2 4 D − D × Motivated by their work, we want to analyze theT stability 1 2 of the charged gravastar in Rastall-Rainbow gravity. For 2GM q1(x) − 1 + this purpose, define a parameter η as follows [65]: − D D2 2m+1 q2(x)D + (m + 1)q3(x)D ′(D) − − 1 × η(D)= P (68) m 2 ̺ (D) 1 q (x)D2 + q (x)D2 +2 − ′ − 2 3 × 1 2 2m+2 2 i The parameter η is interpreted as the squared speed of 1 q2(x)D + q3(x)D sound satisfying 0 η 1 normally. But for the stability − ≤ ≤ h 1 1 of the gravastar, the restriction may not be satisfied on the 2GM q (x) 2 − 1 + 1 (64) surface layer. We have plotted η vs radius D in fig. 18 − − D D2 # and seen that η decreases as radius D increases. Also we 9

140 -0.84 120 -0.86

100 -0.88

80 L L -0.90 D H D H w Ρ 60 -0.92

40 -0.94

20 -0.96

0 -0.98 0 1 2 3 4 5 0 1 2 3 4 5 D D

Fig. 10 represents the plot of surface energy density ̺(D) vs Fig. 14 represents the plot of equation of state parameter radius D. w(D) vs radius D.

ΡHDL wHDL

-21 0.03 -22 10 10 0.02 -23 8 0.01 -24 8 -25 6 6 1 E 1 E 2 4 2 4 3 3 Λ 2 Λ 4 4 2

5 5

Fig. 11 represents the plot of surface energy density ̺(D) vs Fig. 15 represents the plot of equation of state parameter Rastall parameter λ and test particle’s charge E. w(D) vs Rastall parameter λ and test particle’s charge E.

0 80 000 -50

-100 60 000

L -150 D H

M 40 000 P -200

-250 20 000 -300

-350 0 0 1 2 3 4 5 0 1 2 3 4 5 D D

Fig. 12 represents the plot of surface pressure P(D) vs radius Fig. 16 represents the plot of the mass of the thin shell M vs D. radius D. PHDL M

-0.2 2.0 10 1.5 10 -0.4 1.0 8 8 -0.6 0.5 6 6 1 E 1 E 2 4 2 4 3 3 Λ Λ 4 2 4 2

5 5

Fig. 13 represents the plot of surface pressure P(D) vs Rastall Fig. 17 represents the plot of the mass of the thin shell M vs parameter λ and test particle’s charge E. Rastall parameter λ and test particle’s charge E. 10

0.9 vacuum model (p = ρ = 0). In the shell region, the fluid 0.8 source follows the EoS p = ρ (ultra-stiff perfect fluid). 0.7 In this region, since we have assumed that the interior

L 0.6

D and exterior regions join together at a place, so the H Η 0.5 intermediate region must be thin shell and the thickness 0.4 of the shell of the gravastar is inﬁnitesimal. So the thin 0.3 1 shell follows with the limit h ( A− ) 1. Under this 0.2 approximation, we have found≡ the analytical≪ solutions 0 1 2 3 4 5 within the thin shell of the gravastar. The physical D quantities like the proper length of the thin shell, energy Fig. 18 represents the plot of η(D) vs radius D. content and entropy inside the thin shell of the charged

ΗHDL gravastar have been computed and we have shown that they are directly proportional to the proper thickness of the shell (ǫ) due to the approximation (ǫ 1). These physical quantities signiﬁcantly depend on≪ the Rastall parameter λ and Rainbow function Σ(x) (which depends on energy of the test particle). From ﬁgures 1 - 9, we 1.0 10 have seen that the proper length of the thin shell, energy 0.8 8 content and entropy inside the thin shell of the charged 0.6 gravastar increase as thickness increases. Also the proper 6 1 E length decreases as radius increases. The energy content 2 4 and entropy inside the thin shell always increase as radius 3 Λ 4 2 increases. Moreover, if Rastall parameter λ and test parti-

5 cle’s charge both increase then the proper length of the thin shell, energyE content and entropy inside the thin shell Fig. 19 represents the plot of η(D) vs Rastall parameter λ and all decrease. According to the Darmois-Israel formalism, E test particle’s charge . we have studied the matching between the surfaces of interior and exterior regions of the charged gravastar and have plotted η vs Rastall parameter λ and test particle’s using the matching conditions, the surface energy density charge in fig. 19 and it decreases as Rastall parameter λ and the surface pressure have been obtained. Also the and testE particle’s charge increase. Figures show that η equation of state parameter on the surface, mass of the always keep positive sign whichE allows the stability of the thin shell have been obtained and the total mass of the gravastar model. charged gravastar have been expressed in terms of the thin shell mass. From figures 10, 14, 16 we have observed that the surface density, equation of state and mass of the VI. DISCUSSIONS thin shell increase as radius increases but from figure 12, we have seen that the surface pressure always decreases as radius increases but keeps negative sign and hence the In this work, we have considered the spherically sym- equation of state keeps negative sign. Also the figures 11, metric stellar system in the contexts of Rastall-Rainbow 15, 17 show that the surface density, equation of state gravity theory in presence of isotropic fluid source with and mass of the thin shell decrease as the increase of electro-magnetic field. The Einstein-Maxwell’s field both Rastall parameter λ and test particle’s charge . equations have been written in the framework of Rastall- But figure 13 shows the surface pressure increases as theE Rainbow gravity. Next we have discussed the geometry of increase of both Rastall parameter λ and test particle’s charged gravastar model. The gravastar consists of three charge . Finally, we have explored the stable regions of regions: interior region, thin shell region and exterior the chargedE gravastar in Rastall-Rainbow gravity in figures region. In the interior region, the gravastar follows the 18 and 19. The figures show that η always decreases but equation of sate (EoS) p = ρ and we have found the keeps in positive sign as the increase of radius, Rastall solutions of all physical quantities− like energy density, parameter λ and test particle’s charge . pressure, electric field, charge density, gravitational mass E and metric coefficients. In the exterior region, we have obtained the exterior Riessner-Nordstrom solution for

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