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Eur. Phys. J. C (2020) 80:429https://doi.org/10.1140/epjc/s10052-020-7993-8
Regular Article - Theoretical Physics
Extended gravitational decoupling (GD) solution for chargedcompact star model
S. K. Mauryaa
Department of Mathematical and Physical Sciences, College of Arts and Science, University of Nizwa, Nizwa, Sultanate of Oman
Received: 4 December 2019 / Accepted: 29 April 2020 / Published online: 16 May 2020© The Author(s) 2020
Abstract In the present article, we have obtained a newsolution for the charged compact star model through thegravitational decoupling (GD) by using a complete geo-metric deformation (CGD) approach (Ovalle, Phys Lett B788:213, 2019). In this approach, the initial decoupled sys-tem is separated into two subsystems namely Einstein–Maxwell’s system and quasi-Einstein system. We solveEinstein–Maxwell’s system by taking well known Tolman–Kuchowicz spacetime geometry in the context of the perfectfluid matter distribution. On the other hand, the second sys-tem introduce the anisotropy inside the matter distributionwhich is solved by taking an EOS in θ components. Theboundary conditions have been derived to determine the con-stants parameter. To support the mathematical and physicalanalysis of the present GD solution, we have plotted all thegraphs for the compact objects PSR J1614-2230, 4U1608-52 and Cen X-3 corresponding to the constant α = 0.001,0.0012 and 0.0014, respectively. Moreover, we also stud-ied the equilibrium and stability of the solution. The presentstudy shows that the GD technique is a very significant toolto generalize the solution in a more complex form or onematter distribution to another matter distribution.
1 Introduction
It is a great challenge to find the exact solutions of Einstein’sfield equation (EFE) due to its non-linearity. Last 100 years,many exact solutions for EFE have been obtained in the con-text of perfect fluid matter distribution [1]. But only few ofthem are well behaved that describe astrophysical compactobjects. Therefore, still, the researchers are busy to find theexact solution of EFE in different contexts. Usually, there aretwo methodologies that have great interests among all theresearchers for solving the field equations. Fist procedureis that consider one metric function or decreasing energy
a e-mail: [email protected] (corresponding author)
density in order to solve the system. This approach leadsan equation with other unknowns which is determined byusing the pressure isotropy condition. But this process doesnot provide always an admissible exact solution. In a sec-ond way, we can solve the EFE by specifying an equationof state (EOS) that relates the pressure and density. In thisway, we also did not get an exact solution easily for per-fect fluid matter distribution due to its complicated integralfrom. On the other hand, it is not necessary that the mat-ter distribution should be perfect fluid. This was proved bya theoretical investigation that the stellar configuration maycontain the anisotropic pressure (pr �= pt ) when the mat-ter density is very high, in particular, if it is more than thenuclear density [2–7]. This anistropic pressure introduce ananisotropy inside stellar configuration. Then the solution ofEinstein’s field equation in the presence of anisotropy canbe determined easily as compared to perfect fluid matter dis-tribution. In this connection, Many pioneering works havebeen done by several authors in different context [8–38].The several authors have been also obtained anisotropic andcharged solution using embedding class one condition whichis a much popular methodology from last few years [39–60].The presence of the electric charge in the static charged fluidcompact object creates a repulsive force that averts the grav-itational collapse [61,62]. Therefore, we can avoid the sin-gularities by introducing the electric field inside the matterdistribution. This repulsive force, due to the electric charge,balances the equilibrium for the dust distribution against thegravitational collapse. Recently, several authors have dis-covered the solution of the Einstein–Maxwell equations ina different framework which describe strange quark stars,charged black hole, and other astrophysical compact objects[63–71]. Varela et al. [72] studied self-gravitating, charged,anisotropic fluid in solving Einstein–Maxwell equations. Indoing so they considered Krori and Barua’s [73] metricpotentials and linear or nonlinear equation of state. Arbanilet al. [74] studied a class of charged compact fluid spheres
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obeying the polytropic equation of state. They studied theOppenheimer–Volkoff limit, Buchdahl limit for the chargedpolytropic spheres, and they found that the limit is extremalas it is a quasi black hole. So, as a result, we can say that thestudy of the charged fluid sphere is very interesting. That’swhy many researchers obtained exact solutions of Einstein–Maxwell field equations by electrifying some well- knownuncharged fluid spheres e.g. Heintzmann’s [75] solution byPant et al. [76], Durgapal IV [77] by Pant and Rajasekhara[78], Durgapal V [77] by Gupta and Maurya [79], Pant I [80]solution by Maurya and Gupta [81]. The positive values ofthe charge parameter K describe completely the interior ofthe charged super-dense astrophysical object. One may con-sult the following works of literature [82–88] to study therelativistic compact stellar system more precisely with anelectric charge. In this connection, recently the anisotropiccharged and uncharged models in modified f (R, T ) theorygravity have been studied [89–92]. On the other hand, last fewyears the gravitational decoupling using minimal geometricdeformation (MGD) approach has attracted the researchers.In this directions, few well recognize solutions of the Ein-stein field equations have been investigated in the frame-work of charged, uncharged and anisotropic matter distri-bution by using MGD [93–113]. This MGD technique wasdeveloped to deform Schwarzschild space-time [114,115]into the Randall–Sundrum framework [116,117]. Ovelle etal. [118] and Contreras and Bargueno [119] have extendedthe black hole in 3+1 (Schwarzschild outer space-time) andBTZ black hole in 2+1 dimensions using the MGD, respec-tively. The technique for determining an anisotropic solutionfrom any isotropic solution and inverse problem in the blackhole context using the MGD is investigated by Contrerasand his collaborators [120,121]. Recently, Ovelle [122] hasproposed the extended case of MGD to solve the Einstein–Maxwell equations. In this connection, the extended grav-itational decoupling in 2 + 1 dimensional space-time withcosmological context was discovered in [128]. This MGDapproach has been also applied in other areas like Klein–Gordon scalar fields as an extra matter content [125], in cloudof strings [124], an extension of isotropic coordinates [126],ultra-compact Schwarzschild star or gravastar [127], as wellas extended Durgapal isotropic model in the context of classone space-time for charged anisotropic matter distribution[129]. More recently MGD approach was used to discoverthe higher dimensional compact objects [130] and extent inthe context of Lovelock [131] and modified f (R, T ) [91]gravity theories as well as in cosmological problems [132].Moreover, through this GD approach, we can extend any wellbehaved isotropic solution (as discussed above) of the Ein-stein field equation into anisotropic or charged, as well asboth anisotropic-charged domains by adding extra source θi j
in the original energy-momentum tensor Ti j or by definingthe action for total energy tensor. The presence of this extrasource θi j in the system can create an anisotropy or electriccharge inside the matter distribution.
In the present article, we have applied the CGD approachby defining the modified action for total energy momentumtensor for the charged matter distribution which includesthe extra source θi j . In this situation, the field equationsfor complete or original system will contain ten unknownfunctions for charged perfect fluid matter distribution. Bytaking this into account, in the present article we solvethese field equations using gravitational decoupling via com-plete geometric deformation (CGD) approach by transform-ing the metric potentials as: ξ �→ ν = ξ + α h(r) andμ �→ e−λ = μ + α f (r). In this way, we arrive two sys-tems of equations namely Einstein–Maxwell’s system forperfect fluid and quasi-Einstein system corresponding to thesource Ti j and θi j , respectively. However, we suppose thisθi j will produce anisotropy inside the stellar structure. Then,we need to solve the Einstein–Maxwell’s system of equa-tions for anisotropic matter distribution. To do this, first,we solve Einstein-Maxwell’s system by using well knownTolman–Kuchowicz gravitational potentials in the context ofperfect fluid matter distribution. Then for finding the solu-tion of a second system (quasi-Einstein system) we use the anequation of state (EOS), that relates the θ components, andspecific physically motivated ansatz for deformation func-tion f (r). In this way, we obtain a well-behaved expressionfor the deformation function h(r). After solving both sys-tems individually, we combine both solutions and discussedseveral physical analysis in order to get the viability of thismethodology.
The structure of the article is organized as follows: TheSect. 2 is the Einstein’s-Maxwell field equations for the gravi-tationally decoupled system. This section is divided into threesections namely as: (2.1) the action for gravitational decou-pling system, (2.2) basic stellar equation for the decoupledsystem, and (2.3) the gravitational decoupling approach andstellar field equations for Ti j and θi j . The matching condi-tions are performed to determine all the constants in Sect. 3.In the Sect. 4, we obtain a gravitational decoupling solutionfor the charged anisotropic compact star model by using CGDapproach. The physical analysis has been done in Sect. 5where we discussed regularity condition and causality con-dition in order to put the bound on the constants. The Sect. 6contains the detailed discussion on the compactness and red-shift. We have performed the equilibrium and stability anal-ysis of the solution in Sect. 7. The physical feasibilty of theenergy conditions have been verified in Sect. 8. The finalSect. 9 has been made for discussion and conclusion.
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2 The Einstein–Maxwell field equations forgravitationally decoupled system
2.1 The action for gravitational decoupled system
The modified action for gravitational decoupled system canbe defined by introducing an extra Lagrangian density for anextra source as [122],
S = SEH + Sθ =∫ [
R
16π+ LM + Le + Lθ
]√−g d4x .
(1)
where LM and Le denote the Lagrangian for matter field andelectromagnetic field, respectively, while Lθ is a Lagrangiandensity corresponding to additional source. However, thesymbols R and g has their usual meanings.
Now we define Ti j as a energy-momentum tensor forLagrangian matter field LM which can be written as,
Ti j = − 2√−g
δ(√−gLM
)δgi j
, (2)
As we know that matter Lagrangian LM depends on only thecomponents of metric tensor gi j and not on their derivatives,so we obtain
Ti j = −2 ∂(LM )
∂gi j+ gi jLM , (3)
On the other hand, we denoted Ei j and θi j as electromagneticfield tensor and extra source corresponding to Lagrangiandensity Le and Lθ , respectively, which can be given by,
Ei j = −gi j Le + 2∂Le
∂gi j, (4)
θμν = −2δLθ
δgμν+ gμνLθ . (5)
Now by varying the action (1) with respect to the metrictensor gi j we obtain the general equations of motion for thedecoupled charged system as,
Gi j = −8π(T toti j + Ei j
), (6)
where,
T toti j = Ti j + θi j . (7)
Here,Gi j denotes the Einstein tensor and the relativistic unitsare to be taken G = c = 1. We define Ti j (corresponding toperfect fluid matter distribution) and Ei j as,
Ti j = (ρ + p)uμuν − pgi j (8)
Ei j = 1
4π
(−Fm
μ Fν m + 1
4gi j Fγ n F
γ n)
. (9)
Here ρ and p represent the matter density and pressure forcharged matter distribution whileuν is a covariant componentfor the 4-velocity which fulfil uμuμ = 1 and uμ∇μuμ = 0.
On the other hand, the anti-symmetric electromagnetic fieldtensor Fμν satisfies the Maxwell’s field equations,
∇v
[(−g)1/2 Fi j
]= 4π (−g)1/2 jμ, (10)
Fμν,γ + Fνγ,μ + Fγμ,ν = 0, (11)
where j i , denotes the electromagnetic four current vector, isgiven by,
J i = σ√g44
dxi
dx4 = σ vi , (12)
where σ denotes the charge density and defined as σ =eν/2 J 0(r). Moreover, the non-vanishing component for thefour-current static fluid matter distribution is J 4. Therefore,the four-current component acts only along radial directiondue to the spherical symmetry. Then the corresponding non-zero components for electromagnetic field tensor are F01 andF10 that are related as F01 = −F10. Then the electric fieldalong the radial direction r can be defined by these compo-nents F01 and F10 only.
2.2 Basic stellar equation for decoupled system
In order to describe interior spacetime for the sphericallysymmetric static stellar system, we take the following lineelement as,
ds2 = −eλ(r)dr2 − r2(dθ2 − sin2 θ dφ2) + eν(r) dt2,
(13)
where the metric potentials ν and λ are depend on the radialcoordinate r only. Then the Einstein’s field equations for thestatic spherically symmetric spacetime (13) correspondingto decoupling system (6) can be written as,
e−λ
8π
(λ′
r− 1
r2 + eλ
r2
)= (T 0
0 )tot + E00 , (14)
−e−λ
8π
(ν′
r+ 1
r2 − eλ
r2
)+ 1
r2 = (T 11 )tot + E1
1 , (15)
− e−λ
32π
(2ν′′ + ν′2 + 2
ν′ − λ′
r− ν′λ′
)= (T 2
2 )tot + E22 .
(16)
where,
(T 00 )tot = T 0
0 + θ00 , (17)
(T 11 )tot = T 1
1 + θ11 , (18)
(T 22 )tot = T 2
2 + θ22 . (19)
Then the linear combination of equations (14) -(16) yieldsthe following conservation equation,
− ν′2
(T 0
0 − T 11 + E0
0 − E11
)+
(T 1
1 + E11
)′ − 2
r
(T 2
2 − T 11
)
− 2
r
(E2
2 − E11
)− ν′
2
(θ0
0 − θ11
)+
(θ1
1
)′ − 2
r
(θ2
2 − θ11
)= 0, (20)
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However for the spherically symmetric line element (13), thenon-vanishing anti-symmetric electric field components F01
and F10 can be given in terms of the electric charge as,
F01 = −F10 = q
r2 e−(ν+λ)/2 (21)
where q(r) denotes the electric charge contained within thecompact star of radius r . By using relativistic Gauss law, theelectric charge q(r) as well as electric field E can be definedas,
q(r) = 4 π
∫ r
0σ r2 eλ/2dr = r2
√−F10 F10, (22)
E2 = −F10 F10 = q2
r4 . (23)
Then the components for T ij and Ei
j can be expressed as,
T 00 = ρ(r), T 1
1 = −p(r), T 22 = −p(r), (24)
E00 = E1
1 = −E22 = 1
8π
q2(r)
r4 . (25)
Then the conservation equation (20) will take the followingform as,
−p′ − ν′
2(ρ + p) + qq ′
4 πr4 − ν′
2
(θ0
0 − θ11
)+ (θ1
1 )′
−2
r
(θ2
2 − θ11
)= 0. (26)
On the other hand, it is clearly noted that the inclusion of newsource θi j in the system introduced the anisotropy inside thematter distribution, if θ2
2 �= θ11 . In this situation, the total
energy tensor T tot, given in Eq. (7), can be described as,
(T 00 )tot = ρtot(r) = ρ(r) + θ0
0 (r), (27)
−(T 11 )tot = ptot
r (r) = p(r) − θ11 (r), (28)
−(T 22 )tot = ptot
t (r) = p(r) − θ22 (r). (29)
Hence, the anisotropy factor is given as,
�(r) = ptott (r) − ptot
r (r) = θ11 (r) − θ2
2 (r) (30)
2.3 The gravitational decoupling approach and stellar fieldequations corresponding to Ti j (for charge matterdistribution) and extra source θi j
In this section, we will see that how the extended gravitationaldecoupling converts the field equations (14)–(16) in two sep-arate systems which are known as the Einstein–Maxwell sys-tem and quasi-Einstein system, respectively. As the previousdiscussion, we take the Einstein–Maxwell system with theperfect matter distribution for Ti j while the quasi-Einsteinsystem corresponding to extra source θi j . Now we apply theOvalle [122] transformation in metric potentials as,
ξ �→ ν = ξ + α h(r) (31)
μ �→ e−λ = μ + α f (r). (32)
where h(r) and f (r) represents the geometric deformationfunctions corresponding to the temporal and radial metriccomponent. The coupling constant α is a real number. More-over, the above transformation is the extended case of min-imal geometric deformation (MGD) which is called a com-plete geometric deformation (CGD) or extended geometricdeformation along with both radial and temporal componentsof the line element. By substituting the deformed metric func-tions (31) and (32) in the field equations of decoupled system(14)–(16) with Eqs. (24) and (25), we arrive at the followingset of equations:
(I) The Einstein’s equations of the charged perfect fluid mat-ter distribution for energy momentum tensor Ti j are givenas (corresponds to α = 0),
1 − μ
r2 − μ′
r= 8πρ + q2
r4 , (33)
μ − 1
r2 + μξ ′
r= 8πp − q2
r4 , (34)
μ
(ξ ′′
2+ ξ ′2
4+ ξ ′
2r
)+
(ξ ′μ′
4+ μ′
2r
)= 8πp + q2
r4 .
(35)
along the conservation equation,
− p′ − ν′
2(ρ + p) + qq ′
4 πr4 = 0. (36)
The solution of the field equations (33)–(35), satisfiesconservation equation (36) that can describe the internalstructure of the compact object for the charged perfectfluid matter distribution. The corresponding solution canbe given by the following line element,
ds2 = −μ dr2 − r2(dθ2 − sin2 θ dφ2) + eξ dt2 (37)
where the interior mass function m0 is a mass functionof the charged matter distribution for the standard GRexpressions (33)–(35), which can be given as
2m0
r≡1−μ+q2
r2 ≡8π
r
∫ r
0
(ρ+ q2
8π r2
)r2 dr+q2
r2 .
(38)
(II) Now let us go on the parameter α to see the effects ofthe extra source θi j on the charged perfect fluid solution{ξ, μ ρ, p, }. For this purpose we write the field equationsfor the quasi-Einstein system associated with the sourceθi j as ,
−α
(f ′r
+ f
r2
)= 8 π θ0
0 , (39)
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−α f
(ν′r
+ 1
r2
)− αF1 = 8 π θ1
1 , (40)
−α f
2
(ν′′ + ν′2
2+ ν′
r
)− α f ′
2
(ν′2
+ 1
r
)= 8 π θ2
2 + αF2.
(41)
and the linear combination of quasi-Einstein equa-tions (39)–(41) yields the following conservation equa-tion,
− ν′
2
(θ0
0 −θ11
)+
(θ1
1
)′ −2
r
(θ2
2 − θ11
)= h′
2(p+ρ).
(42)
where, F1 and F2 are given as,
F1 = μ h′r
, F2 = μ
4
(2 h′′ + α h′ 2 + 2 h′
r+ 2 ξ ′ h′
)+ μ′ h′
4.
It is clearly noted that the presence of θ -sector in thesystem will introduce the anisotropy inside the stellarstructure.
3 Matching conditions for the stellar structure
The matching condition is a critical part in the study of thestellar distributions at the surface of the star (r = R) betweeninterior (r < R) and exterior (r > R) spacetime geometries.In the present study, the interior stellar geometry is given bythe extended geometric deformation line element,
ds2 = −(
1 − 2 m̃(r)
r+ q2
r2
)−1
dr2 − r2(dθ2 − sin2 θ dφ2)
+eξ(r)eα h(r) dt2, (43)
where m̃(r) is internal mass of the stellar structure fortotal energy tensor T tot
i j . Now the inner metric (43) should besmoothly matched with an exterior spacetime geometry. Inhis regard, Ovalle [122] has proposed that the exterior space-time for the extended gravitational decoupling correspondingto “Maxwell version” of the vacuum solution Ti j = 0 can begiven by the well-known Reissner–Nordstrom solution as,
ds2 = −(
1 − 2M̃
r+ Q2
r2
)−1
dr2 − r2(dθ2 − sin2 θ dφ2)
+(
1 − 2M̃
r+ Q2
r2
)dt2, (44)
where m̃(R) = M̃ and q(R) = Q is the total mass andtotal charge for the compact object of the radius R, respec-tively. For smooth joining we apply the Israel–Darmois junc-tion conditions procedure [133,134] to match the inner man-ifold M− with the external one M+ at the boundary �.
The procedure of joining both space-time at the boundary isknown as the continuity of the first and second fundamentalforms across the surface �. The first fundamental form saysthat the intrinsic geometry described by the metric tensor gi jinduced by M− and M+ on the interface meets
g−t t |r=R = g+
t t |r=R, (45)
g−rr |r=R = g+
rr |r=R . (46)
By writing the first fundamental in the explicit form as,
e−λ(R) =(
1 − 2M̃
R+ Q2
R2
), (47)
eν(R) =(
1 − 2M̃
R+ Q2
R2
). (48)
While the continuity of the second fundamental can bedescribed as,
[Gi j rj ]� = 0 (49)
here r j is a unit vector. Using Eqs. (6) and (49) we can find
[T toti j r j ]� = 0, (50)
which gives,
[ptotr
]�
=[pr − θ1
1 (r)]�
= 0 (51)
Now we will explain by simple way, for the standard GR casethe Ti j = 0 is the vacuum solution for the region r > R.Here r = R is the boundary of the stellar structure. But theexterior spacetime (r > R) may not be a vacuum anymoredue to presence of new fields which is coming from the θ
sector. This extra field introduce the anisotropy inside theself-gravitating system. Then the above boundary condition(51) will take the following final form,
pr (R) − (θ11 )−(R) = −(θ1
1 )+(R) (52)
where pr (R) = p−r (R). The condition given by Eq. (52)
is called the general expression for the second fundamentalform connected with the Einstein field equations describedby Eq. (6). Now we substitute the value of (θ1
1 )−(R) for theinterior geometry from Eq. (40) into the condition (52). Thenthe second fundamental form (52) can be written as,
pr (R) + α
8π
[f (R)
(ν′(R)
R+ 1
R2
)− μ(R) h′(R)
R
]
= −(θ11 )+(R), (53)
here ν′(R) ≡ ∂r ν−|r=R . To obtain (θ11 )+(R), we use the
Eqs. (38) and (48) for the outer geometry in Eq. (53) whichgives,
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pr (R) + α f (R)
8π
(ν′(R)
R+ 1
R2
)− α μ(R) h′(R)
8πR
= α f ∗(R)
8π
⎡⎢⎢⎣ 2
R2
(M̃
R−Q2
R2
)1(
1 − 2M̃R + Q2
R2
) + 1
R2
⎤⎥⎥⎦
−α(h∗(R)
)′
8πR
(1 − 2M0
R+Q2
R2
), (54)
where, f ∗(R) and h∗(R) are geometric deformation func-tions for the outer Reissner–Nordstrom solution (44) due tothe extra source θμν . It is a important remark that if the exte-rior solution is given by the Reissner-Nordstrom solution(44) then we must have f ∗(r) = 0 and h∗(r) = 0. Nowfrom Eq. (54) we get,
pr (R) + α f (R)
8π
(ν′(R)
R+ 1
R2
)− μ(R) h′(R)
8πR= 0 (55)
which can also read as,
ptotr (R) = pr (R) − θ1
1 (R) = 0. (56)
The conditions (47), (48) and (56) are the necessary andsufficient conditions for determining the arbitrary constantsinvolve in the system.
4 Gravitational decoupling solution
As we see that system of equations (33)–(35) and (39)–(41)contains ten unknowns. Therefore, in order to solve the firstsystem of equations (33)–(35) we consider the seed space-time (37) corresponding to well-known Tolman–Kuchowiczsolution as,
μ = 1
(1 + a r2 + b r4), (57)
ξ = B r2 + C. (58)
here, a, b, B are the parameters of dimension l−2, l−4, andl−2, respectively while C is a dimensionless constant. As weobserve that both μ and ξ are non-singular and well behavedthroughout within the stellar model. These forms of the grav-itational potential yield a physically viable solution. From theEqs. (33) and (34) together with pressure isotropy condition(using in Eqs. (34) and (35)), we obtain the expressions forρ, p and q2/r4 as,
ρ = a2 r2 + b r2 (5 + b r4) + a (3 + 2 b r4)
8π (1 + a r2 + b r4)2 − q2
8π r4 , (59)
p = (2 B − a + b r2)
8π (1 + a r2 + b r4)+ q2
8π r4 , (60)
q2
r4 = r2[a2 + B2 + b2r4 − a(B − 2br2 − B2r2) + ψ1(r)]2 (1 + ar2 + br4)2 . (61)
where, ψ1(r) = b(B2r4 − 1 − 2Br2).
Now we focus on the second system of equations (39)–(41). As we see that this system involes five unknown func-tion namely θ0
0 , θ11 , θ2
2 , and two deformation functions f (r)and h(r). For solving of this system, we propose a linearequation of state (EOS) in θ along with the one deformationfunction f (r). But It is important to note that the choice off (r) should depend upon the following points:
1. For positive α together with positive and increasing func-tion f (r), the growth of μ(r) must be faster than the thedeformation function α. f (r) in order to preserve eλ(r)
and mass function to be positive and increasing.2. For positive α, and negative decreasing function f (r),
the metric function eλ(r) and mass function will increasethroughout automtically.
Based on above discussions we take,
θ00 = β θ1
1 + γ, (62)
f (r) = −a r2 (1 + a r2 + b r4). (63)
Now using (39), (40) together with EOS (62) and (63), weobtain other deformation function h(r) as,
h(r) = r2 [h1(r) + h2(r) + h3(r) + h4(r)]60 α β
(64)
with
h1(r) = 30[aα(β − 3) + γ ] + 15a[2aα(β − 4)
+2Bα β + γ ]r2,
h2(r) = 10 [a3α (β − 5) + 2abα(β − 5)
+4a2Bαβ + b γ ]r4,
h3(r) = 15 a α [a b (β − 6) + a2 B β + 2 b B β] r6,
h4(r) = 6 a b α [b (β − 7) + 4 a B β] r8 + 10a b2B α βr10.
Now we have completely determined the deformationfunctions f (r) and h(r), which are physically viable (Fig. 1).Then the gravitational decoupling solution for the totalenergy momentum tensor T tot
i j can be given by followingline element,
ds2 = −(1 + a r2 + b r4)
1 + α (1 + a r2 + b r4) f (r)dr2 − r2(dθ2
+ sin2 θ dφ2) + eB r2+C eα h(r) dt2. (65)
Hence, the explicit form of the deformed gravitational poten-tials read as,
eλ(r) = (1 + a r2 + b r4)
1 + α (1 + a r2 + b r4) f (r), (66)
eν(r) = eB r2+C eα h(r). (67)
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where the interior mass function m̃(r) for decoupled sys-tem (14)–(16) can be given by,
2m̃(r)
r= 1−e−λ(r)+q2
r2 =8π
r
∫ r
0
(ρtot + q2
8π r2
)r2 dr
+q2
r2 . (68)
Then from Eqs. (27), (38) and (68), we get the followingrelation,
m̃(r) ≡ m0(r) − α r
2f (r) = 4π
∫ r
0ρ r2 dr − α r
2f (r).
(69)
It is noted that the masses m̃(r) and m0 will be equal anddescribe the mass function for the standard GR case whenα = 0.
Now the expressions for the components of θ -sector aredetermined as,
θ00 (r) = a α (3 + 5 a r2 + 7 b r4)
8 π, (70)
θ11 (r) = 5 a2 α r2 + a α (3 + 7 b r4) − γ
8 π β, (71)
θ22 (r)=θ11(r)+θ12(r)+θ13(r)+a2r2[θ14(r)+θ15(r)]
−32 π α β2 (1 + ar2 + br4).
(72)
where the used coefficients in θ22 (r) are given in appendix
due to long expressions.By substituting the Eqs. (59)–(61) together with Eqs. (70)–
(72) into Eqs. (27)–(29), we find the expressions for the totaldensity (ρtot) and total radial pressure (ptot
r ) and total tangen-tial pressure (ptot
t ) as,
ρtot(r) = a(6 + Br2 + 2br4 − B2r4) + ψ2(r)
2(1 + ar2 + br4)2 + θ00 (r), (73)
ptotr (r) = a (3Br2 − 2br4 + B2r4 − 2) + ψ3(r)
2(1 + ar2 + br4)2 − θ11 (r), (74)
ptott (r) = a (3Br2 − 2br4 + B2r4 − 2) + ψ3(r)
2(1 + ar2 + br4)2 − θ22 (r) (75)
and anisotropy factor � can be calculated by the for-mula: �(r) = ptot
t (r) − ptotr (r).
where,
ψ2(r) = a2r2+r2[b2r4+b(11+2Br2 − B2r4) − B2],ψ3(r) = B2(r2+br6)−a2r2 + 2B(2+br4) − br2(3 + br4).
5 Physical analysis
To be a regular model, it requires that the spacetime shouldbe free from any mathematical and geometrical as well asphysical singularity. For this purpose, we need to check thevariation of the deformed metric potentials within the stellar
interior. From Eqs. (66) and (67) we obtain: eλ(0) = 1 andeν(0) = eC . The variation of the metric potentials is presentedin Fig. 1 (bottom right) which shows metric potentials areregular and positive throughout the model.
On the other hand the central values of total pressure (ptot0 )
and total density (ρtot0 ) can be given ,
(ptotr
)r=0 = (
ptott
)r=0 = 2 B β − a(3α + β) + γ
8 π β> 0, (76)
⇒ B >a (3 α + β) − γ
2 β, (77)
(ρtotr
)r=0 = 3 a (1 + α)
8 π> 0 ⇒ a is positive. (78)
Now, the solution should satisfy the Zeldovich condition i.e.the ratio of pressure-density should be less than unity, then(ptotr
ρtot
)r=0
=(ptott
ρtot
)r=0
=2 B β − a(3α+β)+γ
3a β (1 + α)≤ 1,
⇒ B ≤ a (3α + 4β + 3α β) − γ
2 β. (79)
In addition to above, the causality condition must satisfyeverywhere within the star i.e. 0 < v2
r < 1 and 0 < v2t < 1.
Then we obtain the central values (v2r )r=0 and (v2
t )r=0 andtogether with causality condition, we obtain
0 < (v2r )r=0 = 5aBβ − (B2 − 3b)β − a2(3β − 10α)
[B2 − 11b − aB − a2(10α − 11)] β< 1, (80)
0 < (v2t )r=0 = b(6αβ2 − γ 2) − 2 B α β (B β + 2 γ ) − δ1
2 α β2 [B2 − 11 b − a B + a2(11 − 10 α)] < 1. (81)
Then the inequality (80) gives,
5a
2−
√12 b β + a2 (40 α + 13 β)
(2√
β< B
<3a
√β − √
28bβ + a2 [20 α (1 + β) − 19β]2
√β
, (82)
while inequality (81) yields,
2 a α β [α (6 − 4β) + 5 β] − 4 α β γ − �1
4 α β2 < B
<a α β(3α + 3β − 2α β) − α β γ − �2
2α β2 . (83)
Where �1 and �2 are given in the appendix. Now combineall the inequalities (77), (79), (82) and (83), we obtain aninequality that restricts B as,
a (3 α + β) − γ
2 β< B
<3a
√β − √
28bβ + a2 [20 α (1 + β) − 19β]2
√β
. (84)
Also, the variation of total density (ρtot) and total radial andtangential pressures (ptot
r and ptott ) are shown in Fig. 2, which
shows that those are positive and decreasing away from cen-tre. However the maximum values attain at the centre of the
123
429 Page 8 of 16 Eur. Phys. J. C (2020) 80 :429
0.00.20.40.60.81.01.21.41.61.82.02.22.4
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
egrahccirtcele
r/R
α = 0.0010α = 0.0012α = 0.0014
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
|f(r
)|
r/R
α = 0.0010α = 0.0012α = 0.0014
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
h(r)
r/R
α = 0.0010
α = 0.0012
α = 0.0014
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
noitcnufcirte
m
r/R
Fig. 1 The behavior of electric charge, q(r) (top left), deformationfunctions, | f (r)| (top right) and h(r) (bottom left), and bottom rightfigure for gravitational potentials eλ (solid lines) and eν (dashed lines)have been plotted against the radial coordinate r/R of different compactobjects as PSR J1614-2230 (black curve for α = 0.001), 4U1608-52
(green curve for α = 0.0012), and Cen X-3 (red curve for α = 0.0014).For plotting of these figures for each object, we have used the val-ues of free parameters as: a = 0.00509, b = 0.00002, β = 1.8 andγ = 0.0001
compact object. The numerical values of the physical quan-tities and constant parameters are given in Table 1.1
6 Mass-radius ratio and surface redshift
It is required to discuss the maximum limit of the mass-radius ratio to describe the compactness of the stellar model.In the case of isotropic matter distribution, the maximumlimit of the mass-radius ratio was proposed by Buchdahl’s[139] in the framework of perfect fluid having decreasingenergy density towards to boundary. This maximum limitfor mass-radius is given as,
M
R≤ 4
9, (85)
1 we have used the geometrical units throughout the paper expect thenumerical values as mentioned in table. Moreover, I have mentionedthe required units of the used parameters in the tables.
where m(R) = M describes the total mass of the object inperfect fluid matter distribution, while the R represents theradius of the model which is obtained by taking the pressureto be zero on the surface. Moreover, the presence of an electriccharge in the solution modifies Buchdahl’s limit. In this case,Andreasson [140], and Bohmer and Harko [141] have pro-vided the maximum and minimum limit for the mass-radiusratio, respectively as,
Q2(18R2 + Q2
)2R2
(12R2 + Q2
)≤M0
R≤2R2 + 3Q2 + 2R
√R2 + 3Q2
9R2 .
(86)
where m0(R) = M0 is the total mass of the compact objectfor the charged perfect fluid matter distribution. It is notedthat the mass M0 present in the Eq. (86) is not equal as thetotal mass appears in (85). This can be defined as,
M0 = m0(R)=4π
∫ R
0r2ρ(r)dr+1
2
∫ R
0
q2(r)
r2 dr+Q2(R)
2R
123
Eur. Phys. J. C (2020) 80 :429 Page 9 of 16 429
0.00000
0.00001
0.00002
0.00003
0.00004
0.00005
0.00006
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
p r)tot(
r/R
α = 0.0010
α = 0.0012
α = 0.0014
0.00000
0.00001
0.00002
0.00003
0.00004
0.00005
0.00006
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
p t(to
t)
r/R
α = 0.0010α = 0.0012α = 0.0014
0.00030
0.00035
0.00040
0.00045
0.00050
0.00055
0.00060
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
ρ)tot(
r/R
α = 0.0010α = 0.0012α = 0.0014
0.0000000
0.0000001
0.0000002
0.0000003
0.0000004
0.0000005
0.0000006
0.0000007
0.0000008
0.0000009
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
rotcafyportosi na
r/R
α = 0.0010
α = 0.0012
α = 0.0014
Fig. 2 The behavior of total radial pressure ptotr (top left), total tangen-
tial pressure ptott (top right), total energy density ρtot
r (bottom left), andanisotropy factor � (bottom right) verses radial coordinate r/R. The
used numerical values for plotting these figures along with the samedescription of the curve for each compact object are same as used inFig. (1)
Table 1 The numerical values of mass, radius, central pressure, cen-tral density, surface density, electric charge at surface and constantsB and C of the different compact objects, namely PSRJ1614-2230
[135,136], 4U1608-52 [136,137], Cen X-3 [136,138] for fixed value ofa = 0.00509km−2, b = 0.00002km−4, β = 1.8, and γ = 0.0001 withdifferent values of α
α Compactstar
Mass M̃/M PredictedradiusR (km)
Centralpressureptotc (dyne/cm2)
Centraldensityρtotc (gm/cm3)
Surfacedensityρtots (gm/cm3)
Charge q atsurface inCoulomb
B (in km) C
0.0010 PSR J1614-2230 1.97 11.0143 7.8786 × 1034 8.2082 × 1014 4.8357 × 1014 2.8578 × 1020C 0.00333 -1.0595
0.0012 4U1608-52 1.74 10.514 7.4522 × 1034 8.2098 × 1015 5.1265 × 1014 2.4628 × 1020C 0.0033 -0.96175
0.0014 Cen X-3 1.49 9.922 6.9168 × 1034 8.2115 × 1014 5.4688 × 1014 2.0361 × 1020C 0.0053 -0.7738
Table 2 Comparative study of lower bound, Mass-radius ratio, upper bound, compactness (u = Meff/R) and surface red-shift of the star fordifferent values of α
α Compact star MassM̃/M
RadiusR (km)
LowerboundQ2 (18R2+Q2)
2R2 (12R2+Q2)
Mass-radiusratio( M0
R )Mass-radiusratio( M̃R )
Upperbound2R2+3Q2+2R
√R2+3Q2
9R2
Surface redshift zs
0.0010 PSR J1614-2230 1.97 9.62 0.03710 0.17847 0.26383 0.47689 0.38425
0.0012 4U1608-52 1.74 10.514 0.03025 0.16508 0.24411 0.47097 0.34576
0.0014 Cen X-3 1.49 9.922 0.02322 0.14949 0.22109 0.46487 0.30321
123
429 Page 10 of 16 Eur. Phys. J. C (2020) 80 :429
= R
2
[1 − μ(R)+Q2(R)
R2
]. (87)
On the other hand, the gravitational mass (appears inEq. (85)) and effective mass both will be same in the contextof perfect fluid or anisotropic fluid matter distribution, whichcan written as,
M = m(R) = 4π
∫ R
0r2ρ(r)dr= R
2[1−μ(R)] = [
M]
eff.
(88)
But the above situation is not same for the charged matterdistribution. In this case, the effective mass for charged matterdistribution can be given as,
[M0
]eff = [
m0(R)]
eff = 4π
∫ R
0
(ρ + q2
8π r4
)r2dr
= R
2[1 − μ(R)] . (89)
Now from the equations (87), (88) and (89), we observedthe followings: (i) the total mass M0 for charged stellar objectwill be more than the total mass M of the compact object cor-responding to the prefect or anisotropic fluid matter distribu-tion, (ii) the effective mass [M0]eff of the charged compactstellar model will be same as the total mass M in contextof prefect fluid or anisotropic fluid matter distribution. Ofcourse, the gravitational mass for the electrically chargedstellar object has more value than the uncharged perfect fluidstellar model. Moreover, the same scenario will happen forthe present gravitational decoupling models2. Now our aimis to see that whether the compactness i.e. mass-radius ratioin the presence of gravitational decoupling will take morevalue than the without gravitational decoupling, and also itwill go beyond to the above standard bounds or not?. Then
from the Table 2, we see that the mass-radius ratio ( M̃R ) forGD model is also lying within the range given in Eq. (86). ButIt has more value than the mass-radius ratio ( M0
R ) in absenceof anisotropy (means when α = 0). Then it can be concludedthat the presence of θ -sector i.e. anistropy, in the system canproduce more compact objects. On the other hand, we wouldlike to mention here that the effective mass plays an impor-tant role to define the upper bound of the surface redshift zsfor the compact object. As we have already discussed thatthe mass-radius ratio gives a significant bound to observe the
2 we refer the following notations which have been used inside the textto represent the mass function for different matter distribution:
(i) m(R) = M −→ for perfect fluid matter distribution without grav-itational decoupling i.e. α = 0.
(ii) m0(R) = M0 −→ for charged perfect fluid matter distributionwithout gravitational decoupling i.e. α = 0.
(iii) m̃(R) = M̃ −→ for anisotropic charged fluid matter distributionby gravitational decoupling i.e. α �= 0.
0.25
0.30
0.35
0.40
0.45
0.50
0.55
0.60
0.65
0.70
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
tfihsder
r/R
α = 0.0010α = 0.0012α = 0.0014
Fig. 3 The variation of gravitational redshift (z) verses radial coordi-nate r/R. The same values are taken for plotting of this figure as inFig. 2
surface redshift zs value. This bound can be determined bythe following formula,
zs = (1 − 2u)−1/2 − 1 = eλ(R)/2 − 1,
where u ≡ M̃eff
R= 1
R
([M0
]eff − α R
2f (R)
). (90)
On the other hand the gravitational redshift inside the com-pact object for GD solution can be obtained as,
z = e−ν(r)/2 − 1 =√e−(B r2+C) e−α h(r) − 1. (91)
We have shown the behavior of the gravitational redshiftinside the star in Fig. 3. From this figure, we can see that thegravitational redshift is maximum at the centre and decreas-ing outward. Table 2 demonstrated the numerical values ofthe surface redshift for different values of coupling constantα.
7 Equilibrium and stability for the gravitationaldecoupling model
It is very important to investigate the stable equilibriumfor the gravitational decoupling solution. For doing thisinvestigation, we need to find the generalized Tolman-Oppenheimer-Volkoff (TOV) equation for the charged aniso-tropic matter distribution in the presence of extra gravita-tional source θi j . This generalized TOV equation can beobtained from the the conservation equation of T tot
i j i.e. by
taking covariant derivative of T toti j to be zero, which yields:
∇μT toti j = 0. The explicit form of this conservation can be
expressed as,
− (ptotr )′ − ν′
2
(ρtot+ptot
r
)+ qq ′
4 πr4 −2
r
(θ2
2 −θ11
)= 0.
(92)
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Eur. Phys. J. C (2020) 80 :429 Page 11 of 16 429
-0.000015
-0.000010
-0.000005
0.000000
0.000005
0.000010
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
ecroF r/R
Fig. 4 The behavior of different forces: Fg (solid lines), Fh (longdashed-dot), Fe (long dashed), and Fa (small dashed-dot) verses radialcoordinate r/R. The description of the figure is as follows: (i) red colorcurves corresponding to the star PSR J1614-2230 for α = 0.001; (ii)green color curves corresponding to the star 4U1608-52 for α = 0.0012and (iii) green color curves corresponding to the star Cen X-3 forα = 0.0014. The numerical values of free parameters are: a = 0.00509,b = 0.00002, β = 1.8 and γ = 0.0001.
which is same as Eq. (26). Also, the above Eq. (92) can bedivided in different forces whose linear combination will bal-ance the system and achieve a stable equilibrium of the solu-tion. These forces can be written as: (i). gravitational force:Fg = − ν′
2 (ρtot+ ptotr ), (ii). hydrostatic force: Fh = −(ptot
r )′,(iii). electric force: Fe = 2 q q ′
8 π r4 , and (iv). anisotropic force:
Fa = 2r (θ2
2 − θ11 ).
To find the variations of these forces we plot the Fig. 4 formodified TOV equation (92) for different values of α. FromFig. 4 we observe that gravitational force (solid lines) andhydrostatic force (long dashed-dot lines), and like anisotropicforce (small dashed-dot lines) are increasing and achieve itsmaximum value at the point within the stellar interior andthen start decreases in each case. while other forces likeelectric force (long dashed lines) are monotonically increas-ing for throughout within the stellar interior. Another impor-tant point is that the electric force (Fe) plays a major roleto balance the system near to surface of the objects whileansitropic force introduces a less impact on the system. Aswe see that the gravitational forces can be balanced by jointaction of all other forces like hydrostatic force, electric forceand anisotropic force such that Fg + Fh + Fe + Fa = 0. Inthis way, we achieved the stable equilibrium of the obtainedeach stellar model.
After analyzing the hydrostatic equilibrium under differ-ent forces, it is also required to check the stability analysisof the stellar model. To check this analysis, we use Abreu’scriteria [38] which has been initiated by Herrera’s crackingconcept [142]. According to the Abreu’s criterion, the stellarmodel will be stable if the subliminal radial and tangentialsound speeds satisfy the following inequalities (Fig. 5),
0.00.10.20.30.40.50.60.70.80.91.0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
V r2
r/R
α = 0.0010α = 0.0012α = 0.0014
0.00.10.20.30.40.50.60.70.80.91.0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1V t
2
r/R
α = 0.0010
α = 0.0012
α = 0.0014
Fig. 5 The left panel has been plotted for radial speed (v2r ) and right
panel for the tangential speed (v2t ) verses radial coordinate r/R of dif-
ferent compact objects as PSR J1614-2230 (black curve for α = 0.001),4U1608-52 (green curve for α = 0.0012), and Cen X-3 (red curve forα = 0.0014). The numerical values of the free parameters a, b, β andγ are same as used in Fig. 4
0 ≤ |v2t − v2
r | ≤ 1
={
(v2t − v2
r ) ∈ [0, 1], Potentially unstable,(v2
t − v2r ) ∈ [−1, 0] Potentially stable
}. (93)
In order to verify these above inequalities, first, we needto check whether the stellar model is satisfying causalitycondition or not?. Then from Fig. 5, we see that the radialand tangential subliminal speed of sounds are satisfying thecausality conditions i.e. v2
r < 1 (top figure) and v2t < 1
(bottom figure) throughout within stellar object (where thespeed of light is taken as unity i.e. c = 1). But we mention aninteresting point that both radial and tangential velocities ofsound are decreasing monotonically and radial velocity (v2
r )is always greater than the tangential velocity (v2
r ) through-out within the stellar compact object for each chosen valueof α, which can be analyzed from Fig. (5). After verifyingthe causality condition, we check the stability condition ofthe stellar compact model through the Eq. (93). Form Fig. 6,we observe that stability factors v2
t − v2r (dashed lines) and
v2t − v2
r (solid lines) are lying within the intervals [−1, 0]and [0, 1]. This implies that the radial velocity dominatesthe tangential velocity everywhere inside the stellar interior.
123
429 Page 12 of 16 Eur. Phys. J. C (2020) 80 :429
-0.009
-0.007
-0.005
-0.003
-0.001
0.001
0.003
0.005
0.007
0.009
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
rotcafytili bat s r/R
Fig. 6 The trend of stability factors corresponding to v2r − v2
t (solidlines) and v2
t − v2r (dashed lines) verses the radial coordinate r/R. The
description of the compact objects are as follows: PSR J1614-2230(black curve) for α = 0.001, 4U1608-52 (green curve) for α = 0.0012,and Cen X-3 (red curve) for α = 0.0014. The numerical values of thefree parameters a, b, β and γ are same as used in Fig. 4
Also, there is no cracking within the star. Therefore, we con-clude that the obtained self-gravitating charged compact starmodels are stable.
8 Energy conditions
It is very important to check the feasibility of some inequal-ities corresponding to the stress-energy tensor. For this pur-pose, we study these inequalities so-called energy condi-tions for representing physically realistic matter configura-tion. The respective energy conditions viz., the null energycondition (NEC), strong energy condition (SEC) and weakenergy condition (WEC) are defined as
WEC : Tμνlμlν ≥ 0 or ρtot ≥ 0, ρ(tot) + p(tot)
i ≥ 0 (94)
NEC : Tμν tμtν ≥ 0 or ρ(tot) + p(tot)
i ≥ 0 (95)
where Tμνlμ ∈ nonspace-like vector
SEC : Tμνlμlν − 1
2T λ
λ lσ lσ ≥ 0 or ρ(tot) +
∑i
p(tot)i ≥ 0.
(96)
where i ≡ (radial r, transverse t), lμ and tμ are time-like vector and null vector respectively. In order to verify theviable feasibility of the stress-energy tensor, we need to studywhether the compact stellar structure is consistent with theinequalities (94)–(96) or not? For this purpose, we plot theFig. 7 for the above energy conditions and we observe thatour stellar compact objects are consistent with all the energyconditions and hence ratifies that the physical acceptabilityof gravitational decoupling solution for compact objects.
0.00030
0.00040
0.00050
0.00060
0.00070
0.00080
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
snoitidnocygrene
r/R
Fig. 7 The behavior of energy condition verses radial coordinate r/R.The description of the figures as follows: the solid line for ρtot, dottedline for ρtot + ptot
r , small dashed lines for ρtot + ptott , and long dashed-
dotted lines for ρtot + ptott + 2ptot
t for different compact objects PSRJ1614-2230 (black color), 4U1608-52 (green color), and Cen X-3 (redcolor)
9 Discussion and conclusions
In this article, a completely deformed anisotropic chargedfluid solution for the compact star model has been investi-gated by applying gravitational decoupling by means of ageometric deformation approach. To find the gravitationaldecoupling solution for the compact star model, first, wewrite the modified action for the gravitational decouplingsystem. Then we define the equation of motion by vary-ing this action along with the metric tensor gi j . In this way,we arrive the Einstein’s field equation for the coupled sys-tem which corresponds to the total energy-momentum tensorT toti j in the context of spherically symmetric spacetime. The
corresponding Einstein’s field equations for coupled systemwill be solved by using the gravitational decoupling (GD)using a complete geometric deformation approach (CGD)in which both gravitational potentials have bee deformed asξ �→ ν = ξ + α h(r) and μ �→ e−λ = μ + α f (r), asproposed by Ovalle [122]. This GD approach transforms theoriginal Einstein’s field equations for coupled system intotwo individual subsystems corresponding to the new sourcesTi j and θi j . Here we would like to mention that the energytensor Ti j is considered for the charged perfect fluid matterdistribution while θi j introduce the anisotropy in the sys-tem (quasi-Einstein system). To solve the system for Ti j ,we use well-defined spacetime given by Tolman–Kuchowiczand find the solution for the first system, which provides thegravitational potentials μ and ξ and electric field E2. Nowstill five unknowns namely θ0
0 , θ11 and θ2
2 , h(r) and f (r) areremaining in order to describe the complete structure of thestellar structure. Since it is clear that we have three indepen-dent equations for determining these five unknown whichimplies that we have two degree of freedom. To find this,we solve the quasi-Einstein system of Eqs. (39)–(41) cor-
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Eur. Phys. J. C (2020) 80 :429 Page 13 of 16 429
responding to θ -sector by specifying the linear equation ofstate (EOS), θ0
0 = β θ11 + γ , and well motivated ansatz for
f (r). After solving of this EOS, we obtain another deforma-tion function h(r). In this process, we achieved a completedeformed charged anisotropic solution for the compact starmodel. Moreover, we use the well known Isrial–Dormoisjunction condition for determining the constant parameters.The physical properties of the solution are described throughthe graphical analysis. We would like to mention that all theplots have been made for the same values of a = 0.00509,b = 0.00002, β = 1.8 and γ = 0.0001 with different α. Inthis way we obtained three different compact stars: (i) PSRJ1614-2230 for α = 0.001, R = 11.0143km, (ii) 4U1608-52for α = 0.0012 and R = 10.514 km, and (iii) Cen X-3 withα = 0.0014 and R = 9.922km. Here, we would like to men-tion that Demorest et al. [135] have predicted the radius cor-responding to the differential neutron star EOS. They foundthat the radius of the compact objects with 1.97 solar masslies between 11 and 15 km. But for strange quark matter (SSEOS), the radius will be less for this compact object withthe same 1.97 solar mass. Later on, Gangopadhyay and hiscollaborators [136] have predicted the radii corresponding to12 different stars and fitted the refined mass measurement of12 pulsars using strange star equation of state (SS EOS) andthey found that the star PSR J1614-2230 with solar mass 1.97solar has the radius, approx. 9.69 km. In the present paper,By varying α we have predicted the radius of the realisticcompact objects, as above, for fixed mass value. Then weobserve the following points: if α increases then we will getless compact object as well lower masses stars. But for highervalues of α ≈ 0.15 either causality will violates or crackingwill appears in the system.
The Fig. 1 shows the behavior of the deformation func-tions h(r), | f (r)|, electric charge q(r) and gravitationalpotentials eν(r) and eλ(r). From this figure, we observe thatthe functions | f (r)| (as f (r) is negative and monotonicdecreasing throughout the star) and h(r) are zero at thecentre and increasing towards the boundary of the stellarobject and the same scenario happens in q(r) also. Theamount of charge on the surface of the star in Coulombunit as: (i) Q = 2.8578 × 1020C for PSR J1614-2230 withα = 0.0001, (ii) Q = 2.4628 × 1020C for 4U1608-52 withα = 0.0012, and (iii) Q = 2.0361 × 1020C for Cen X-3with α = 0.0014. However, the eν and eλ both are posi-tive and increasing monotonically for all values of α andfree from singularity. The above well-defined regular gravi-tational potentials describe a well behaved physically viablegravitational decoupling solution for compact objects. On theother hand the Fig. 2 also shows the behavior of total radialand tangential pressures, (ptot
r and (ptott ), total density (ρtot),
and the anisotropy (�) against the radial coordinate r/R. Aswe can see in this Fig. 2 that the total pressures and den-sity are maximum at centre and decreasing monotonically
away from centre, while the anisotropy is zero at centre andincreasing monotonically towards the boundary. The radialpressure is vanishes of the boundary of star which decidethe size of the compact model i.e. radius R while tangen-tial pressure is not. We determined the total mass (M̃), andsurface redshift (zs) of each obtained compact object whichare the most important physical features of the model. It wasproposed that the mass M̃ has larger value as compared tothe total mass M0, which means that anisotropy introducemore massive objects. We have presented the numerical val-
ues of the compactness( M̃R
)and
(M0R
)for fixed values of
a = 0.00509, b = 0.00002, β = 1.8 and γ = 0.0001for the different compact objects as PSR J1614-2230 forα = 0.001, R = 11.0143 km, 4U1608-52 for α = 0.1,R = 10.5146 km, and Cen X-3 for α = 0.15, R = 9.922 km.Apart from this we have also calculated the lower and upperbound for the same values ofa,b,α,β andγ which shows thatthe each compactness factor (mass-radius ratio) lies betweenthe lower and upper bound. Through the compactness, wehave evaluated the surface redshift of the each different com-pact star model which are given as follows: (1) zs = 0.38425for PSR J1614-2230, (2) zs = 0.34576 for 4U1608-52, (3)zs = 0.30321 forCen X-3. These obtained values for surfaceredshift are compatible with the values proposed by Ivanov[63] and Bowers and Liang [8]. Moreover, the variation ofgravitational redshift within the charged ansitropic compactstellar models are shown in Fig. 4. The gravitational red-shift is maximum at centre and decreasing towards the sur-face boundary, and attains minimum at surface. On the otherhand, we verified the stable equilibrium of the gravitationaldecoupling solution via different forces. To do so, we needto study the modified Tolman–Oppenheimer–Volkoff (TOV)equation for anisotropic charged matter distribution. We havepresented distributions of all forces Fg , Fh , Fa , and Fe inFig. 4. From this Fig. 4, we found that the anisotroic forceFα and electric force Fe acts along an outward direction.However, the combined impact of the hydrostatic force Fh ,anisotropic force Fa and electric force Fe will balance thegravitational force Fg such that Fg + Fh + Fa + Fe = 0,which yields the stable equilibrium for the matter distributionT toti j . Therefore, our obtained GD solution is in the equilib-
rium stage. Finally, we discussed the causality and stabilityof the GD solution. The Fig. 5 shows that the speed of soundsis decreasing throughout the stellar interior and less than thespeed of light. Moreover, the radial velocity is always greaterthan the tangential one which can be predicted from Fig. 5. Onthe other hand, from Fig. 6, we also see that the values of thestability factors v2
t −v2r (dashed lines) and v2
r −v2t (solid lines)
belong to the intervals [−1, 0] and [0, 1], respectively. Also,The anisotropic model satisfies all the energy conditions (seeFig. 7). Finally, we would like to mention that the obtainedcharged anisotropic solution satisfies all the mathematical
123
429 Page 14 of 16 Eur. Phys. J. C (2020) 80 :429
and physical requirements which shows that the extendedgravitational decoupling by means of a complete geometricdeformation approach is very effective and significant toolfor generalizing or finding the new solution of the Einstein’sequations.
Acknowledgements S. K. Maurya acknowledges that this work is car-ried out under TRC project-BFP/RGP/CBS/19/099 of the Sultanate ofOman.
Data Availability Statement This manuscript has no associated data orthe data will not be deposited. [Authors’ comment: There is no externaldata associated with this manuscript.]
Open Access This article is licensed under a Creative Commons Attri-bution 4.0 International License, which permits use, sharing, adaptation,distribution and reproduction in any medium or format, as long as yougive appropriate credit to the original author(s) and the source, pro-vide a link to the Creative Commons licence, and indicate if changeswere made. The images or other third party material in this articleare included in the article’s Creative Commons licence, unless indi-cated otherwise in a credit line to the material. If material is notincluded in the article’s Creative Commons licence and your intendeduse is not permitted by statutory regulation or exceeds the permit-ted use, you will need to obtain permission directly from the copy-right holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.Funded by SCOAP3.
The used coefficients in the above expressions
θ11(r) = a6bα2r10(−5 + β + 2Bβr2)2 + 4a5bα2r8
(−5 + β + 2Bβr2)(−4 − 6br4
+β(1 + 2Br2)(1 + br4)) + γ (bγ (r + br5)2
+4αβ(1 + 2br4 + B(r2 + br6)))
+2a4bαr6[γ r2(−5 + β + 2Bβr2)
+α{47 + 146br4 + 107b2r8
+3β2(1 + 2Br2)2(1 + br4)2
−12β(1 + 2Br2)(2 + 5br4 + 3b2r8)}],θ12(r) = 2a3αr4[αβ(−25 + β − 10Br2 + 8Bβr2
+2B2βr4) + 2b4αr12(42 − 13β(1 + 2Br2)
+(β + 2Bβr2)2) + 2b3αr8{88 − 33β(1 + 2Br2)
+3(β + 2Bβr2)2} + b{γ r2(−13 + β(3 + 6Br2))
+2α(12 − 7β(1 + 2Br2) + (β + 2Bβr2)2)}+b2r4{γ r2(−17 + β(3 + 6Br2))
+2α(58 − 27β(1 + 2Br2) + 3(β + 2Bβr2)2)}],θ13(r) = 2a[bγ 2r4(1 + br4)
+αγ r2{β(3 + 2Br2) + b4r12(−7 + β + 2Bβr2)
+b(−3 + β + 2Bβr2)
+b3r8(−17 + β(3 + 6Br2)) + b2r4(−13
+β(3 + 6Br2))} + 2α2β{−3 + b(−27 + β)r4
+b2(−28 + β)r8
+βr4(B + bBr4)2
+Br2(1 + br4)(−3 − 7br4 + β(2 + 6br4))}],θ14(r) = b5α2r16(−7 + β + 2Bβr2)2
+2α2β(−29 + β − 16Br2
+12Bβr2 + 4B2βr4) + 4b2αr4(−5 + β
+2Bβr2)[3γ r2 + α(−3 + β + 2Bβr2)]+4b4α2r12[35 − 12β(1 + 2Br2)
+(β+2Bβr2)2]+2b3αr8[γ r2(−19+β(3+6Br2))
+α(71 − 30β(1 + 2Br2) + 3(β + 2Bβr2)2)],θ15(r) = b[γ 2r4 + 2αγ r2(−11 + β(3 + 6Br2))
+α2{9 − 2β(3 + 79r4
+6B(r2 + 4r6)) + β2(1 + 6r4 + 4B(r2 + 10r6)
+4B2(r4 + 2r8))}],�1 =
√�11 − 8 α β2 (a2 α �12 + �13),
�2 =√
α β2(28 b α β2 + a2 α �21 − �22),
�11 = [2aαβ(−5β + α(4β − 6)) + 4αβγ ]2,
�12 = b α (β − 3)2 + 2β[α(β − 23) + 3β],�13 = 2 a α [b (β − 3)+β] γ+b(6αβ2−γ )2,
�21 = α2(3−2β)2−bα (β − 3)2−19 β2+2α β(32 + 3β)
�22 = 2 a α [α (3−2β)+b(β − 3)+4β] γ − b γ 2+α γ 2.
with,
δ1 = a2α[bα(−3 + β)2 + 2β(α(−23 + β) + 3β)]+2 a α [B β(−5β + α(−6 + 4β))+(b(−3+β)+β)γ ].
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