aa r X i v : . [ phy s i c s . g e n - ph ] F e b Construction of Charged CylindricalGravastar-like Structures
Z. Yousaf ∗ Department of Mathematics, University of the Punjab,Quaid-i-Azam Campus, Lahore-54590, Pakistan
Abstract
In this work, we extend the work of gravastars to analyze the role of electromagneticfield in f ( R, T ) gravity. We consider the irrotational cylindrically symmetric geometryand established the f ( R, T ) field equations and conservation laws. After consideringcharged exterior geometry, the mathematical quantities for evaluating Israel junctionconditions are being calculated. The mass of the gravastar-like cylindrical structure iscalculated through the equations of motion at the hypersurface in the presence of anelectromagnetic field. The behavior of electric charge on the length of the thin shell,energy content, and entropy of gravastar is being studied graphically. We concludedthat charge has an important role in the length of the thin shell, energy content, andentropy of gravastar.
Keywords:
Gravitation, Cylindrically symmetric spacetime, Isotropic pressure.
Gravitation is perhaps the basic interaction one can experience most effectively in dailyactivities. General relativity, along with the quantum theory, could be seen as the rise ofmodern physics. The consequences of relativity must be brought into consideration whenexamining galactic weak gravitational force. There are eloquent relativistic sources, such aswhite dwarfs, neutron stars, and black holes, in which these effects could have significantresults. As a matter of interest, it becomes absolutely essential to follow observationally ∗ [email protected] f ( G ) theory, where G is a Gauss-Bonnet term [4], f ( G, T ) [5, 6, 7] (where T isthe trace of energy-momentum tensor), f ( R, (cid:3) R, T ) [8, 9] (where (cid:3) is the de Alembert’soperator), and f ( R, T, R γδ T γδ ) [10, 11, 12, 13, 14], etc. Several articles and reports exist inthe literature on the discussion of dark energy [15, 16] and MGTs [17, 18, 19, 20, 21, 22,23, 24, 25] as an elaboration for cosmic acceleration. The simplest alternative version of GRis f ( R ) theory, attained by displacing the Ricci scalar R in the EH action with its genericfunction. As a a plausible explanation of cosmic inflation, this theory has been source ofgreat attraction among many researchers due to the inclusion of quadratic R terms [26] in theEH action. Harko et al. [27] laid out the grounds for f ( R ) gravity and the concept of f ( R, T )( T is introduced due to quantum effects or exotic imperfect content), is introduced in whichnon-minimal coupling between geometry and matter was established. They formulated thecorresponding equations of motion describing certain astrophysical perspectives.Houndjo [28] attempted to reconstruct cosmological models in f ( R, T ) theory and arguedthat his models might feasibly explain accelerated and matter-dominated cosmic histories.After employing the perturbation method on the mathematical power-law models, Baffou et al. [29] studied cosmic instability problems and determined their viability requirements.Sahoo et al. [30] analyzed phases of spatially homogeneous cosmos in f ( R, T ) gravity. Re-cently, Moraes et al. [31] inspected the presence of stellar bodies in our accelerating universeand managed to find some viability constraints in R + χT gravity, where χ is a constant.Yousaf et al. [32] examined the role of f ( R, T ) gravity after subjecting the initially homo-geneous compact structure into the inhomogeneous one and found a relatively slow rate ofcollapse due to f ( R, T ) correction. They also provided relations connecting the histories ofinflationary and late-time acceleration of the universe. Bhatti et al. [33, 34, 35, 36] alsoworked in f ( R, T ) cosmology for different astrophysical models.Gravitational collapse is amongst the most remarkable phenomena of the stellar dynam-ics. The endpoint of gravitational collapse may result in black holes, neutron stars or whitedwarfs. Although there exists significant works that indicate that the collapse of theoret-ically, acceptable matter contents may result in the formation of naked singularity (NS).Herrera et al. [37] examined NS formation in the spherical relativistic matter distributionwith the help of Weyl tensor. Virbhadra et al. [38, 39] provided a theoretical framework due2o which one can study significantly relations between the formation of NS and black holesas a result of celestial collapse.Mazur and Mottola [40] proposed the alternative concept of endpoints in the gravitationalcollapse of supermassive stars. They modified the notion of Bose-Einstein for self-gravitatingsources. They presented the gravitational vacuum star (gravastar), which is alternative tothe black hole. Gravastar has no singularity and event horizon. It is comprised of threespecific layers interior region, middle thin shell and exterior region. The middle shell is verythin with the finite range r < r < r . Here r = r is the radius of interior and r = r + ǫ is theradius of the exterior region of gravastar, ǫ indicates the thickness of the middle shell, whichis very very small. According to the Mazur and Mottola model, these layers of gravastarhas three specific EoS. ρ + p = 0 is the EoS for interior region, ρ − p = 0 for thin shell and ρ = p = 0 for the exterior region.Rahaman et al. [41] discussed charge region of gravastar under (2+1)-dimensional space-time. They analyzed various theoretical feasible features, like length, entropy, energy for thespherical configuration. Pani et al. [42] considered irrotational thin-shell gravastar modelsby taking Schwarzschild and de Sitter metric as exterior and interior geometries, respectively.They performed linearized stability analysis in order to view the existence of such structuresin the cosmos. Cecilia et al. [43] presented a detailed analysis in order to study gravastarson the dynamical properties of fast rotating those compact objects that do not possess ahorizon. They declared that not all gravastars which rotate are unstable and the origin ofgravastars is still highly speculative.Cattoen et al. [44] presented the usual concept of a gravastar and indicated that thesecould be treated as an alternative to the black hole. They considered a family of mathe-matical models characterizing with continuous pressure. They calculated the role of somephysical variables, like anisotropy pressure, minimization of the anisotropic region, etc inthe formation of gravastars at the boundary surface. Cardoso et al. [45] considered theproperties of rotating gravastars with the help of the perturbation scheme and estimatedtheir stability within the slow-rotation approximation. They examined the instability effectson those objects which do not have an event horizon like Kerr black hole, gravastars, andboson stars. They concluded that when these ultra-compact objects undergo rapid spinning,then a strong ergoregion instability may appear in their dynamics.Ghosh et al. [46] examined the possibility of the existence of gravastars in higher dimen-sional spacetime and inferred that few important indications about the formation of suchcelestial structure with or without electric charge. Das et al. [47] examined a family ofsingularity-free gravastar models in f ( R, T ) gravity and claimed that f ( R, T ) gravity couldlikely to host such structures. Shamir and Mustaq [48] extended these results and exam-ine the role of f ( G, T ) gravity on the possible formation of gravastars. Recently, Yousaf et al. [49] presented non-singular spherically symmetric gravastar model after evaluating3iability conditions in the presence of Maxwell- f ( R, T ) corrections. Bhatti [50] describedthe modeling of cylindrical gravastars-like structures in the field of GR.This paper aims to analyze the role of Maxwell- f ( R, T ) corrections on the possible mod-eling of gravastars-like cylindrical structures, a viable alternative to the final stage of grav-itational collapse. The paper consists of the following sections. In the coming section, webriefly elaborate the basic framework of f ( R, T ) theory and established the field equationof modified theory. Section is devoted to computing modified field equations and mass ofthe gravastar with an EoS. In section , we shall examine the effects of electromagnetic fieldon the viable matching conditions between exterior and interior geometries. In section , wedetermine a few characteristics in cylindrical gravastar-like structures. In section , we willdiscuss the role of electric charge on some physical valid properties of gravastar. f ( R, T ) Theory and Charged Cylindrical System
One can modify the EH action for f ( R, T ) gravity as under S = 116 π Z f ( R, T ) √− g d x + Z L m √− g d x, (1)where L m describes the Lagrangian density for an ordinary fluid, g and T are the tracesof metric and the energy momentum tensor, respectively. The energy momentum tensorthrough L m can be defined as follows T ζη = − √− g δ ( √− g L m ) δg ζη = g ζη L m − ∂ L m δg ζη . (2)Varying the action of f ( R, T ) theory with respect to metric tensor provides δS = 116 π Z (cid:20) f R δR + f T δTδg ζη δg ζη − g ζη f δg ζη + 16 π √− g δ ( √− g L m ) δg ζη (cid:21) √− g d x, (3)where f R ( R, T ) and f T ( R, T ) indicate the partial derivative with respect to Ricci scalar andtrace of energy momentum tensor respectively. In Eq.(3), one can find the value of δR asunder δR = δ ( g ζη R ζη ) = R ζη δg ζη + g ζη ( ∇ γ δ Γ γζη − ∇ η δ Γ γζγ ) , (4)in which δ Γ γζη = 12 g γβ ( ∇ ζ δg ηβ + ∇ η δg βζ − ∇ β δg ζη ) . (5)4sing Eqs.(5) in Eq.(4), we get δR = R ζη δg ζη + g ζη (cid:3) δg ζη − ∇ ζ ∇ η δg ζη , (6)where (cid:3) ≡ ∇ α ∇ α . Equation (6) eventually makes Eq.(3) as under δS = 116 π Z (cid:2) f R ( R, T ) R ζη δg ζη + f R ( R, T ) g ζη (cid:3) δg ζη − f R ( R, T ) ∇ ζ ∇ η δg ζη + f T ( R, T ) δ ( g µν T µν ) δg ζη δg ζη − g ζη f ( R, T ) δg ζη + 16 π √− g δ ( √− g L m ) δg ζη (cid:21) √− gd x. (7)After some manipulations, one can write the corresponding equation of motion as follows f R ( R ζη − ∇ ζ ∇ η + g ζη (cid:3) ) + f T ( T ζη + Θ ζη ) = 8 πT ζη + 12 f g ζη , (8)where Θ ζη = g µυ ∂T µυ ∂g ζη . (9)An important point that must be stressed here is that the divergence of an energy-momentum tensor in this theory, unlike GR and f ( R ), is non-zero. It can be given as follows ∇ ζ T ζη = f T ( R, T )8 π − f T ( R, T ) [( T ζη + Θ ζη ) ∇ ζ ln f T ( R, T ) + ∇ ζ (Θ ζη )] . (10)which allows this theory to break both weak and strong equivalence principles. Consequently,one can observe the non-geodesic motion of the test particles. It is possible to recover f ( R )gravity under the constraint f ( T ) = 0.The aim of the present work is to analyze the formation of cylindrical gravastar-likeobjects in f ( R, T ) gravity under the influence of electromagnetic field. For this purpose weassume the following static form of cylindrical metric ds = − H ( r ) dt + K ( r ) dr + r ( dφ + α dz ) , (11)where H ( r ) = q α r − Mαr + q α r and H ( r ) = K ( r ) with M indicates the gravitating massof the object while q is the charge and α is a constant term with the dimensions of inverselength. We then model our source for the above geometry as the locally-isotropic fluidconfiguration, whose mathematical form can be expressed as under T ζη = ( ρ + p ) V ζ V η + pg ζη , (12)5here V ζ is the four velocity, ρ and p denote the density and pressure of the relativisticfluid, respectively, Under comoving coordinate system, the vector V µ satisfies V ζ V ζ = 1relation. To solve the non-linear nature of field equations, as described in [27, 51], we take L m = − p + F for the charged perfect fluid, where F = − F µν F αβ g µν g αβ = − q r , for whichΘ ζη is found to be Θ ζη = − (2 T ζη + pg ζη + F g ζη ). Here F αβ stands for Maxwell tensor. Togive mathematical as well as cosmologically observational consistency to f ( R, T ) theory, thechoice of f ( R, T ) model is of key importance. Therefore, we consider here f ( R, T )= R + 2 χT ,where χ is constant. This simplest case of f ( R, T ) could be fully equivalent to GR, afterrescaling the value of χ in the analysis. In this framework, Eq.(8), turns out to be G ζη = 8 π ( T ζη + E ζη ) + χT ζη g ζη + 2 χ [ T ζη + E ζη + pg ζη + F g ζη ] , (13)where G ζη is an Einstein tensor and E ζη describes an energy momentum tensor for a chargedmatter. It can be given as follows E αβ = 14 π (cid:18) F γα F βγ − F γδ F γδ g αβ (cid:19) . (14)In this background, Eq.(10) takes the following form ∇ ζ T ζη = − χ π + χ ) (cid:2) ∇ ζ ( pg ζη ) + 2 ∇ ζ ( F g ζη ) + g ζη ∇ ζ T (cid:3) . (15)By the substitution of χ = 0 in the above equation, one can simply obtain the GR results.The non-zero components of Einstein tensor for the cylindrically symmetric interior ge-ometry are G = − K r ( ´ Kr − K ) , (16) G = 1 KHr ( ´ Hr + H ) , (17) G = 14 H K r (2 ´´ HHK − H ´ H ´ K − K ´ H r − KH + 2 ´ HKH ) . (18)Making use of Eqs.(11)-(18) in Eq.(13), we obtain the following equations´ Kr − KK = − r [8 πρ − χ ( p − ρ )] − q r (cid:20) χ (cid:18) π − (cid:19)(cid:21) , (19)´ Hr + HHK = − r [8 πp + χ (3 p − ρ )] + q r (cid:20) χ (cid:18) π − (cid:19)(cid:21) , (20)6 H K (2 H ′′ ) HK − H ´ H ´ K − K ´ H r − KH + 2 ´ HKH )= − r [8 πp + χ (3 p − ρ )] + q r (cid:20) χ (cid:18) π − (cid:19)(cid:21) . (21)Here, we noticed that G = α G . For f ( R, T ) theory, the hydrostatic equilibrium condi-tion satisfy Eq.(10) and is defined as´ H H ( ρ + p ) + dpdr − q πr − χ (4 π + χ ) (cid:20)
12 (´ p − ´ ρ ) − q r (cid:21) = 0 . (22)Using Eq.(19), we get1 K = 8 m h + χ ( ρ − p r + q r (cid:20) χ (cid:18) π − (cid:19)(cid:21) , (23)where r represents the radius of interior region of gravastar and m is the correspondinggravitational mass within the interior region. Substituting Eq.(23) in Eq.(22), one canobtain dpdr = q πr − q χr (4 π + χ ) − H ′ H ( ρ + p )[1 + χ π + χ ) (1 − dρdp )] , (24)where H ′ H = Kr [ − πp + χ ( ρ − p )] + Kq r (cid:20) χ (cid:18) π − (cid:19)(cid:21) − r . It is well established that the exterior and interior geometries of gravastars are supportedby particular forms of EoS. We consider that geometric structure of an interior exteriorspacetime is described by p = − ρ relation. One can notice that this EoS describes thecontribution of dark energy in the modeling of the interior region. This is a particular caseof barotropic EoS p = ωρ . Equation (24) could be used to prove the following relation p = − ρ ( constant ) , (25)Using Eq.(25) in Eq.(19), we get1 K = r π + χ ) Y ] − q r (cid:20) χ (cid:18) π − (cid:19)(cid:21) , and K = W H − , (26)7here W is an integration constant. The second of above equation describes a particularrelationship between metric coefficients of our observed spacetime. The gravitational mass M ( D ) for the interior region can be described as M ( D ) = Z r = D πr (cid:18) ρ + q r (cid:19) dr = 2 πD (cid:18) D + q (cid:19) . (27)This equation indicates that the gravitating matter content of the stellar body and its radialdistance have a direct relationship, which can be supposed to be a very important featureof compact relativistic objects. This also shows the main dependence of M on a particularradial coordinate. The above integral occupies the properties of improper integral on setting r = ∞ . But the selection of this radius r is not realistic.Now, we elaborate the geometry of middle thin shell and discuss the effect of electromag-netic charge on the pressure within gravastar. Middle thin shell of gravastar is non-vacuumso it contain have a zero pressure. This zone of the gravastar is formed by the ultrarela-tivistic fluid. The EoS of the thin shell is p = ρ . At zero pressure, the solution of fieldequations is much difficult. To overcome this query, we used some approximation for findingan analytical solution, i.e., 0 < H − ≪
1. Physically, we can say that when the exteriorvacuum metric joins with the interior vacuum metric, they form a thin shell [52]. Using EoS ρ − p = 0 in Eqs.(19) and (20), we obtain ddr (ln K ) = − q r (cid:20) χ (cid:18) π − (cid:19)(cid:21) , (28)On substituting the EoS ρ − p = 0 in Eqs.(19) and (21), we get − r r ´ H H + 1 ! ´ K = K , (29)which upon integrating Eq.(28) yieldsln K = q r (cid:20) χ (cid:18) π − (cid:19)(cid:21) + C. (30)´ HH = e q r [ χ ( π − )] q r (cid:2) χ (cid:0) π − (cid:1)(cid:3) − r . (31)Using EoS p = ρ in Eq.(22), we get ddr (ln p ) = e q r [ χ ( π − )] q r (cid:2) χ (cid:0) π − (cid:1)(cid:3) − r . (32)8igure 1: Plot of p = ρ (km) verses r in the gravastars with χ = 1. Integration of the above equation provides ρ = p = Exp (cid:20) − r + Qr − Q Q e Qr + r (cid:21) , (33)where Q = q (cid:2) χ (cid:0) π − (cid:1)(cid:3) . One can note down that here density has a direct propor-tionality with the charge and radius. This shows that the ultrarelativistic fluid within thecentral shell of the gravastar is relativity denser with huge charge at the outer regions thanthat found in the inner regions of the stellar body. Further, when we move out from centralpoint of the region, the more dense with high charged fluid can be noticed. The exterior re-gion should obey EoS, p = ρ = 0 and can be defined through an exterior charged spacetime,for details pleasee see [50]. In this section, we formulate the junction condition by matching interior and exterior re-gions and establish some relations that could allow the corresponding geometries to matchsmoothly on the hypersurface. It is worth notable that the formulation of gravastar is basedon three regions. Let r and r is the radii of interior and exterior regions. We will useDarmois and Israel [52, 53] junction conditions to formulate the smooth matching betweeninterior and exterior surfaces. We will find out surface stress-energy tensor S nm with the help9f Darmois and Israel formulation. Lanczos equation is defined as [54, 55, 56, 57] S nm = − π ( ℓ nm − δ nm ℓ ii ) , (34)where ℓ mn = L + mn − L − mn , (+) and (-) signs indicate the interior and exterior surfaces.The second fundamental [49, 58, 59, 60] form (extrinsic curvature) is defined byΩ ± αβ = ∂x µ ∂ξ α ∂x ν ∂ξ β ∇ µ n ν , (35)= − n ± ω (cid:20) ∂ x ω ∂ξ α ∂ξ β + Γ ωµν ∂x µ ∂ξ α ∂x ν ∂ξ β (cid:21) Σ , (36)where ξ α and x µ can be determined through the coordinates of hypersurface (Σ) and man-ifold. Further, the vector n ± ω corresponds to the double-sided normals on the surface ofcylinder which can be defined as n ± ω = ± (cid:12)(cid:12)(cid:12) g γδ ∂g ( r ) ∂x γ ∂g ( r ) ∂x δ (cid:12)(cid:12)(cid:12) ∂g ( r ) ∂x ω , (37)with timelike condition n ξ n ξ = 1. The values of the energy-momentum tensor at the bound-ary surface turn out to be S nm = diag [ ̺, − ϑ, − ϑ, − ϑ ]. Here ̺ is the surface density and ϑ isthe pressure on the surface. The expressions of ϑ and ̺ are given below ̺ = − πD [ p g ( r )] + − , (38) ϑ = − ϑ π " g ( r ) ′ p g ( r ) + − . (39)Using Eqs.(38) and (39), we get ̺ = − πD "r D α − MDα + 4 q D α − r D π + χ ) ρ − Qr , (40)and ϑ = 18 πD D α − q D α q D α − MDα + q D α − ρ D (2 π + χ ) + Qr − Qr q D (2 π + χ ) ρ − Qr . (41)10sing the areal density, we can determine the mass of the thin shell as m s = 4 πD ̺ = D "r D α − MDα + 4 q D α − r D π + χ ) ρ − Qr , (42)where M = − m s α D − D α π + χ ) ρ + DαQ r + D α q Dα + m s α r D π + χ ) ρ − Qr . (43)is the total amount of quantity of matter within the charged gravastar. In this section, we observe the effect of electromagnetic field on some physical valid charac-teristic of gravastar.
Let us consider that r = D is the radius of interior and r = D + ǫ is the radius of exteriorregion of gravastar. According to Mazur and Mottola [40], proper length of the middle shellcan be determined as ℓ = Z D + ǫD √ K dr = Z D + ǫD p e Q/r + C dr. (44)This equation is not integrable so one can not find its analytic solution. We shell use thenumerical technique and observe the effect of electromagnetic field on proper length of theshell. Figure (2) describes the direct link of the length of the shell with its thickness. It isalso seen from Fig.(2) that as we increase the charge within fluid, then the length of the shellwill be decreased. This indicates that electromagnetic field is trying to lessen down the totallength of the shell of a gravastar. Thus, uncharged gravastars have relatively longer lengthof the shell. 11igure 2:
Plot of ℓ (km) verses ǫ in the gravastars with C = 0 . In the interior region, we used a special combination of EoS given by p = − ρ EoS whichdemonstrates the zone of negative energy that reflects the non-attractive nature of force atthe interior geometry. The energy of the thin shell can be given as ε = Z D + ǫD πρr dr, (45)which turns out to be ε = 4 π Z D + ǫD r Exp (cid:20) − r + Qr − Q Q e Qr + r (cid:21) dr, (46)Again the integration of above expression is not possible so we can not find the analyticsolution. By using numerical method, one can observe the effect of electromagnetic field onthe energy content within the shell. In Fig.(3), the energy ε (km) inside the shell is plottedin relation to the shell thickness ǫ (km) in the presence of electromagnetic field. The plot(3) indicates that less charged fluid has a relatively greater energy within the shell of thegravastars. According to the theory of Mazur and Mottola [40], the interior region of the gravastars hasno disorderness. Now we will find out the entropy by taking r = D and r = D + ǫ radii of12igure 3: Plot of ε (km) verses ǫ in the gravastars with χ = 1. interior and exterior regions, respectively. According to the theory of Mazur and Mottola,entropy within the shell can be calculated using following formula s = Z D + ǫD πr ℜ ( r ) √ K dr, (47)where ℜ ( r ) is the entropy density which can be defined as ℜ ( r ) = (cid:18) αK B ~ (cid:19) (cid:18) P π (cid:19) , K B = ~ = 1 . (48)Using Eqs.(33) and (49) in Eq.(48), the entropy within the shell turns out to be s = (8 πF ) α Z D + ǫD r p e Q/r + C dr, (49)The figure has been plotted in order to analyze the disorderness against the shell thickness.Figure (4) describes zero entropy of the cylindrically symmetric charged gravastar-like rela-tivistic structures at zero thickness of the fluid, which is one of the viable conditions for thesingle condensate phase of the stellar bodies as described by [40].
This paper is devoted to understand the existence of gravastar under cylindrical symmetricmetric in the realm of Maxwell- f ( R, T ) gravity. For this purpose, we consider the cylindrical13igure 4:
Plot of s (km) verses ǫ in the gravastars with χ = 1, F = 0 . C = 0 . geometry and established the subsequent field equations and conservation laws in f ( R, T )gravity. We studied the interior and exterior geometry of gravastar and examined the be-havior of charge on the formulation of pressure and energy density of the gravastar. Weconnected the interior and exterior cylindrical region through some suitable matching con-ditions. Gravastar could be considered as a viable alternative to the black holes, which isthe end state of gravitational collapse. Such an end state can be elaborated with the help ofthree regions containing specific EoS. First is an interior cylindrical region and the second isa middle thin shell and the third is exterior cylindrical region. We also discussed the viablefeatures of gravastar and observed electromagnetic charge effects on it using cylindricallysymmetric spacetime. We determined the mass function of the middle thin shell using inte-rior and exterior geometries. The several salient results found in this work can be describedin the following steps.(i)
Density − P ressure prof ile :The comparison between the two important structural variables, i.e., pressure and densityof the ultrarelativistic matter configuration within the intermediate thin shell are describedin Fig.(1). This escribes a special variation in the profiles of matter variables for charge andradial coordinate. This also describes an abrupt change in the profiles of pressure and energydensity for the low charged fields.(ii)
P roper length of thin shell :The proper length of the thin shell is plotted with respect to its thickness described in Fig.(2),which indicates the progressively increasing profile. Furthermore, an uncharged medium oc-cupies a high value of proper length and vice versa. From the graph (2), we can infer thatif the charge inside the gravastar raises then the length of the thin shell falls and vice versa.(iii)
Energy content :We can see the effects of charge and thickness on the energy content within the thin shell14hrough Fig.(3). It is analyzed that the shell energy has a direct proportion to its thickness.Moreover, the increase in the electric charge will directly increase shell thickness and increaseof electric charge will decrease the amount of energy for the gravastars.(iv)
Entropy :We have illustrated a diagram to investigate the function of electric charge and thicknesson the entropy of the system. Figure (4) indicates the increasing behavior of the entropyby decreasing the contribution of charge and vice versa. Furthermore, the entropy increasesslowly with respect to shell thickness, thereby representing the highest value of thickness athypersurface. Our analysis could be useful to understand the effects of electromagnetic fieldon cylindrically symmetric gravastar-like bodies in f ( R, T ) gravity.
Acknowledgments
This work was supported by National Research Project for Universities (NRPU), HigherEducation Commission, Islamabad under the research project No. 8754/Punjab/NRPU/R&D/HEC/2017.
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