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International Journal of Modern Physics D c World Scientific Publishing Company ⃝

Relativistic fluid spheres with Karmarkar condition

Ksh. Newton Singh Department of Physics, National Defence Academy Khadakwasla, Pune, Maharashtra-411023, India and Department of Mathematics, Jadavpur University, Kolkata-700032, West Bengal, India. [email protected] Nayan Sarkar and Farook Rahaman Department of Mathematics, Jadavpur University, Kolkata-700032, West Bengal, India [email protected] and [email protected] Debabrata Deb Department of Physics, Indian Institute of Engineering Science and Technology, Shibpur, Howrah 711103, West Bengal, India. [email protected] Neeraj Pant Department of Mathematics, National Defence Academy Khadakwasla, Pune, Maharashtra-411023, India. [email protected] Received Day Month Year Revised Day Month Year We present a family of relativistic fluid spheres using Karmarkar criterion along with Pandey-Sharma condition. Ranges of n and m for well-behaved solutions depend on the mass and radius of the compact star model. As an example, we have chosen Vela X-1 (M = 1.77M⊙ , R = 9.56km) for complete analysis of the solutions. For the same star the well behave ranges are 1 < n < 5 and 3 < m < 13. However, beyond these limits either the transversal sound speed becomes imaginary or violates the causality principle. Further, the critical part of our solution is that corresponding to every non-zero positive even value of m yields zero values of all the physical quantities. Keywords: General relativity, Embedded class, Compact stars, Stability criterion, Adiabatic index PACS numbers: 04.20.-q; 04.40.Nr; 04.40.Dg

1. Introduction Recent advances in technology acquiesce a strong constraint on observational astronomy concerning masses and radii of compact stars. Description of some neutron stars such as SAX J1808.4-3658 (a radio pulsar), 4U 1728-34 (an X-ray pulsar), RX 1

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J185635-3754 etc. cannot be incorporated by neutron star model. Therefore, we have to explore for most realistic equation of state (EoS). It is also conjectured that strange quark might be the genuine ground state of hadrons,1, 2 implying the possible existence of strange stars,3–5 quark-diquark stars,6 quark core neutron stars or hybrid stars7 etc. Many researchers have also suggested that the fluid contained inside the compact stars need not be isotropic. Further analysis shows that at such highly dense environment, the fluid distribution are most likely anisotropic due to relativistic nuclear interactions, presence of solid core, type-A superfluid, presence of net electric charge, magnetic field, moment of inertia or rotational motion, phase transition, boson condensation etc. Due to the anistropic in nature the pressure within the compact star divides into two perpendicular components, the radial pressure (pr ) and transverse pressure (pt ), whereas the anisotropic factor is defined as ∆ = 8π(pt − pr ). The anisotropic force is repulsive in nature and it helps to construct a more compact configuration, as proposed by Gokhroo et al.8 There are many alternative methods to discuss the properties of compact stars without knowing the EoS. One of the approaches is known as “core envelop” models where two regions: core and envelop are describe by separate matter distributions. The same approach is devised by many authors successfully.9, 10 In another method, we can ansatz certain physical geometry of interior spacetime, density, EoS, imposing killing symmetry etc. In this article, we are imposing the latter approach by assuming a physically viable spacetime geometry. It is well known that all the spherically symmetric spacetimes are class two solutions i.e. all the spherically symmetric spacetimes can be embedded in 6-dimensional pseudo-Euclidean space. However, if the solutions satisfy the Karmarkar and the Pandey-Sharma conditions they will transform to embedding class one solutions. The same statement was supported by Gupta and Goel11 where they discovered a coordinate transformation that a 4-dimensional Riemannian geometry reduces to 5-dimensional Euclidean geometry. Recently, many researchers have presented physical viable solutions in the background of embedding class one.12–23 2. Einstein field equations for anisotropic fluid The interior space-time is assumed as ds2 = eν(r) dt2 − eλ(r) dr2 − r2 (dθ2 + sin2 θdϕ2 ).

(1)

The Einstein field equations (G = c = 1) are given as 1 Gµν ≡ Rµν − R gµν = 8πTµν . 2

(2)

The stress tensor for anisotropic fluid is defined as Tµν = (ρ + pt )Uµ Uν − pt gµν + (pr − pt )χµ χν

(3)

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3

1.0 700

n = 0 (Blue) n = 5 (Dashed)

0.9

0.8

MeV / fm 3

Metric Potentials

600

0.7

0.6


400

300

0.5

0

500

2

4

6

0

8

2

4

6

8

r ( km )

r ( km )

Fig. 1. Variation of the metric potentials (Left) and density (Right) with the radial coordinate for Vela X-1 (1.77M⊙ , 9.56 km) with a = 0.003/km2 , b = 0.0008/km2 and m = 2n + 3.

50

0.12



0.10

0.08 ωr

30

pt 20

ωr &

pr & pr MeV / fm 3

40

ωt 0.06

0.04

10

0 0

2

4

6

ωr

0.02

pr 8

0.00 0

2

4

6

8

r ( km )

r ( km )

Fig. 2. Variation of pressrues (Left) and equation of state parameters (Right) with radial coordinate for Vela X-1 with the same values used in Fig. 1.

where the symbols have their usual meanings and U µ Uµ = −χµ χµ = 1, U µ χµ = 0. The Einstein field equations for the stress tensor (3) with the interior space-time (1) reduce to λ′ e−λ 1 − e−λ + = 8πρ 2 r r ν ′ e−λ 1 − e−λ − = 8πpr r r2 ( ) e−λ 2ν ′ 2λ′ 2 2ν ′′ + ν ′ − ν ′ λ′ + − = 8πpt 4 r r

(4) (5) (6)

where primes represent differentiation with respect to the radial coordinate r. Using the Eqs. (5) and (6) the expression of anisotropy is

∆ = 8π(pt − pr ) = e−λ

[

] λ′ ν ′ ν ′2 ν ′ + λ′ eλ − 1 ν ′′ − + − + . 2 4 4 2r r2

(7)

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Ksh. Newton Singh et al. 30


1.5

20

m (r) & u (r)

MeV / fm 3

25

15

n = 0 (Blue) n = 5 (Dashed) 1.0

m(r)

10

0.5 5

u(r) 0 0

2

4

6

8

0.0 0

2

4

6

8

r ( km )

r ( km )

Fig. 3. Variation of anisotropy (Left) and mass function (Right) with radial coordinate for Vela X-1 with radial coordinate for Vela X-1 with the same values used in Fig. 1.

2.1. Method of embedding space Following the Kasner method24 let us assume eν = e−λ = 1 − 2m/r in spherical polar coordinates so that (1) reduces to ds2 =

r − 2m 2 2m dt − dr2 − dx2 − dy 2 − dz 2 r r − 2m

where r2 = x2 + y 2 + z 2 . Further, using the transformation √ √ 2m dR = dr or R = 8m(r − 2m), r − 2m

(8)

(9)

the line element(8) reduces to R2 dt2 − dR2 − dx2 − dy 2 − dz 2 . (10) + 16m2 Since the coefficient of the first term in (10) is not perfect differential and hence a 4-D spacetime of the form (1) cannot be embedded in 5-D pseudo Euclidean space. However, introducing another coordinate transformation ∫ √ R sin t R cos t 256m4 X=√ , Y =√ , Z= 1+ 2 dR 2 2 2 2 (R + 16m2 )3 R + 16m R + 16m ds2 =

R2

(11) (10) finally reduces to ds2 = −dx2 − dy 2 − dz 2 + dX 2 + dY 2 − dZ 2 ,

(12)

i.e. a 4-D metric of the form (1) can be embedded in 6-D pseudo Euclidean space. Similarly, Gupta and Goel11 chosen a coordinate transformation ( ) ( ) t t ν/2 ν/2 , z2 = ke sinh , z3 = f (r) (13) z1 = ke cosh k k z 4 = r sin θ cos ϕ, z5 = r sin θ sin ϕ, z6 = r cos θ.

(14)

to transform (1) into ds2 = (dz1 )2 − (dz2 )2 ∓ (dz3 )2 − (dz4 )2 − (dz5 )2 − (dz6 )2

(15)

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Relativistic fluid spheres with Karmarkar condition 700

700



600 - p t MeV / fm 3

600 - p r MeV / fm 3

5

500

400

500

400

300 300

0

2

4

6

0

8

2

4

6

8

r ( km )

r ( km )

Fig. 4. Variation of ρ − pr (Left) and ρ − pt (Right) with radial coordinate for Vela X-1 with radial coordinate for Vela X-1 with the same values used in Fig. 1.

[ ( ) ] with [f ′ (r)]2 = ∓ − eλ − 1 + k 2 eν ν ′2 /4 . Equation (15) also imply that the interior line element (1) can be embedded in 6-D Euclidean space, however, if (dz3 )2 = [f ′ (r)]2 = 0, it can be embed in 5-D Euclidean space i.e. [ ( ) ] f ′ (r)]2 = ∓ − eλ − 1 + k 2 eν ν ′2 /4 = 0 (16) which implies k 2 ′2 ν ν e . (17) 4 The same condition (17) was obtained by Karmarkar26 in the form of components of Riemann tensor as eλ = 1 +

R1414 R2323 = R1212 R3434 + R1224 R1334 .

(18)

27

Even though, Pandey and Sharma pointed out that Karmarkar condition is only the necessary condition to become a class one, discovering the sufficient condition R2323 ̸= 0. Hence, the necessary and sufficient condition to be a class one is to satisfy both Karmarkar and Pandey-Sharma conditions. On integration (17) we get the relationship between ν and λ as ( )2 ∫ √ eν = A + B eλ − 1 dr (19) where A is constants of integration and B = 1/k. By using (19) we can rewrite (7)28, 29 as (for detail see Appendix A.) [ ] [ ′ ν ] ν′ 2 λ′ νe ∆= λ − λ − 1 . 4e r e −1 2rB 2

(20)

3. Generating new embedded class one solutions To solve the above equation (19), we have assumed grr metric as a generalization of Singh et al.30 given as ( )n cr2 ar2 + 1 λ e =1+ (21) m (br2 + 1)

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here a, b and c are constants with dimension of length−2 and are to be determined from the boundary conditions. On integrating (19), we get

[

( ) ]2 √ 2 1−m/2 n/2 2 (br + 1) B c(b − a) m n m abr + a eν = A − ,− ;2 − ; . 2 F1 1 − 2 2 2 a−b (m − 2)bn/2+1 (22)

The trend of eλ and eν are shown in Fig. 1 (Left). On using (21) and (22), we can express ρ, pr , pt and ∆ as

8πρ =

[

( )n−1 c ar2 + 1 n

m 2

( )( )n+1 cr2 br2 + 1 ar2 + 1

(br2 + 1) [cr2 (ar2 + 1) + (br2 + 1) ] ( )m { 2 [ ] + br2 + 1 ar b(3 − 2m)r2 + 2n(br2 + 1) + 3 + b(3 − 2m)r2 ] } +3

(23)

[

][ { } √cr (ar2 + 1)n/2 { )m } ( 2 )n ( 2 2 f1 (r) − f2 (r) 8πpr = + br + 1 r cr ar + 1 m/2 (br2 + 1) { ( )n/2 } ] abr2 + b f3 (r) − Ab(m − 2)r (24) b−a 8πpt = 8πpr + ∆

(25) 2

∆=

r [f4 (r) + f5 (r)] (ar2

[

n

m 2

+ 1) f6 (r) [cr2 (ar2 + 1) + (br2 + 1) ] ] )m { ( ) } ( )( )n+1 ( 2 + br + 1 b amr2 − anr2 + m − an c br2 + 1 ar2 + 1 +

1) (br2

(26)

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700


600


0.8

500 vr

-pr - 2 p t MeV / fm 3

7

0.6

400 0.4

300

0.2

200 0

2

4

6

8

0

2

r ( km )

4

6

8

r ( km )

Fig. 5. Variation of ρ − pr − 2pt (Left) and radial sound speed (Right) with radial coordinate for Vela X-1 with radial coordinate for Vela X-1 with the same values used in Fig. 1.

where ) )n/2 )n/2 ( √ ( 2 A cr ar + 1 abr2 + b − 2Br m/2 b−a (br2 + 1) ( ) ( 2 )( 2 )n m n m abr2 + a Bcr br + 1 ar + 1 ,− ;2 − ; (27) 2 F1 1 − 2 2 2 a−b )n/2 √ ( ) ( B cr ar2 + 1 m abr2 + a m n ,− ;2 − ; (28) 2 F1 1 − m/2−1 2 2 2 a−b (br2 + 1) ( )n/2 [ ] √ abr2 + b Acr(ar2 + 1)n − B cr(ar2 + 1)n/2 (br2 + 1)m/2 b−a b(2 − m) (29) ( ) ) ( 3n/2 Bc3/2 r ar2 + 1 m abr2 + a m n , − ; 2 − ; . (30) F 1 − 2 1 m/2−1 2 2 2 a−b (br2 + 1)

( )m f1 (r) = b(m − 2) br2 + 1 f2 (r) = f3 (r) = f4 (r) =

f5 (r) =

(

The trend of above physical parameters are shown in Figs. 1 (Right), 2 and 3 (Left). The expression for mass-function and compactness parameter can be found as ( )n cr3 ar2 + 1 4πρr dr = m(r) = n m 2 (cr2 (ar2 + 1) + (br2 + 1) ) 0 ( ) n cr2 ar2 + 1 2m(r) = 2 u(r) = n m. r cr (ar2 + 1) + (br2 + 1) ∫

r

2

(31) (32)

The behavior of mass function and compactness parameter are shown in Fig. 3 (Right).

4. Physically acceptability of the new solution The central values of the all the physical parameters must be finite to ensure its physically acceptability. Therefore, the non-singular nature can be seen from the

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following parameters: 8πprc = 8πptc =

( √ B c 2 F1 1 − [ ( b(m − 2)

√ c n m 2 ,−2;2

b b−a

)n/2

−

m a 2 ; a−b

)

− Ab(m − 2)

(

b b−a

)n/2

( √ ) A c − 2B − Bc

( )] m n m a ,− ;2 − ; >0 2 F1 1 − 2 2 2 a−b

(33)

8πρc = 3c > 0 ; ∀ a > 0 (34) prc ≤ 1. (35) ρc Null Energy Condition (NEC), Weak Energy Condition (WEC), Strong Energy Condition (SEC) and Dominant Energy Condition need to satisfy if the matter composition of the stellar fluid are physical matters. ρ ≥ 0; ρ − pr ≥ 0; ρ − pt ≥ 0; ρ − pr − 2pt ≥ 0; ρ ≥ |pr |, |pt |.

(36)

The validity of these energy holds good by the presenting solutions and hence it represents physical matter distribution (Fig. 4 and 5 (Left)). 5. Stability analysis 5.1. Causality and stability The sound velocity inside the compact stars can be determined by using √ √ dpr /dr dpt /dr vr = , vt = . dρ/dr dρ/dr

(37)

Since the sound is propagating through the stellar fluid, it has to be less that light speed (causality condition) or otherwise non-physical stellar fluid. From Figs. 5 (Right) and 6 (Left). Also, for the stability of the system, the stability factor (vt2 − vr2 ) must be in between 0 and −131, 32 which is satisfied by the model, Fig. 6 (Right). 5.2. Adiabatic index and stability The relativistic adiabatic index is also one of the important parameter that affect the stability of any stellar system. It is defined as ρ + pr dpr . (38) Γ= pr dρ For Newtonian limit, any stable configurations will alter its stability by initiating an adiabatic gravitational collapse if Γ ≤ 4/3 and catastrophic if < 4/3.33 According to Chan et al.34 this condition changes for relativistic and/or anisotropic fluid which depends on the nature of anisotropy. For these solutions the adiabatic indices are more than 4/3 with positive anisotropy, Fig. 7 (Left).

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Relativistic fluid spheres with Karmarkar condition 1.0

9

0.00


0.8


-0.02

Stability Factor

-0.04

vt

0.6

0.4

-0.06 -0.08 -0.10

0.2 -0.12

0.0 0

2

4

6

8

-0.14 0

2

4

6

8

r ( km )

r ( km )

Fig. 6. Variation of transverse sound speed (Left) and stability factor (Right) with radial coordinate for Vela X-1 with radial coordinate for Vela X-1 with the same values used in Fig. 1.

5.3. Equation of hydrostatic equilibrium and stability In any stellar fluid systems, the force acting are gravity, hydrostatic pressure and anisotropic force. These forces are to be balance somehow to achieve a system at equilibrium. The balancing force equation is known as TOV-equation. It was generalized by Ponce de Leon35 as −

Mg (ρ + pr ) (λ−ν)/2 dpr 2∆ e − + =0 r2 dr r

(39)

where Mg (r) is the active gravitational mass defined by Tolman-Whittaker mass formula as ∫ r( ) Mg (r) = 4π T00 − T11 − T22 − T33 r2 e(ν+λ)/2 dr. (40) 0

Using the energy-momentum tensor and field equations, the above equation reduce to r Mg (r) = ν ′ e(λ−ν)/2 . (41) 2 Thus equation (39) reduces to Fg + Fh + Fa = 0

(42)

with Fg = −

dpr 2∆ Mg (ρ + pr ) (λ−ν)/2 e ; Fh = − ; Fa = . r2 dr r

(43)

The TOV equation (42) can be represented by a figure showing these forces are balancing to each other Fig. 7 (Right). 5.4. Harrison-Zeldovich-Novikov criterion and stability The stability analysis adopted by Harrison et al.36 and Zeldovich & Novikov37 requires mass should increases with increase in central density or mathematically

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10

10 0.0003

8


0.0002

Fh Fi / km 3

0.0001

Γr

6

Fa 0.0000

Fa + Fg + Fh -0.0001

4 -0.0002

2

Fg

-0.0003 0

0

2

4

6

2

4

6

8

8 r ( km )

r ( km )

Fig. 7. Variation of radial adiabatic index (Left) and TOV-equation (Right) with radial coordinate for Vela X-1 with radial coordinate for Vela X-1 with the same values used in Fig. 1.

0.50


4 0.45

3 0.40 z (r)

M


2

0.35

1 0.30

0 0.000

0.005

0.010

0.015

0.020

0.025

0.030

0

2

ρc ( km -2 )

4

6

8

r ( km )

Fig. 8. Variation of mass with central density (Left) and red-shift (Right) with radial coordinate for Vela X-1 with radial coordinate for Vela X-1 with the same values used in Fig. 1.

dM/dρc > 0. The mass as a function of central density for the solution can be written as ( )n 8πρc R3 aR2 + 1 M (ρc ) = (44) 16πρc R2 (aR2 + 1)n + 6(bR2 + 1)m ( )n ( 2 )m 12πR3 aR2 + 1 bR + 1 dM = [ > 0. (45) n m ]2 dρc 8πR2 ρc (aR2 + 1) + 3 (bR2 + 1) Figure 8 (Left) confirms that the solution holds static stability criterion and hence stable.

6. Matching of interior and exterior space-time The exterior Schwarzschild metric can be written as ( ) ( )−1 2M 2M 2 ds = 1 − dt − 1 − dr2 − r2 (dθ2 + sin2 θdϕ2 ). r r 2

(46)

By matching the interior solution (1) and exterior solution (46) at the boundary

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r = R we get

11

√

√ )1−m/2 B c(b − a)n/2 ( 2 2M =A− 1− bR + 1 R (m − 2)bn/2+1 ( ) m n m abR2 + a ,− ;2 − ; 2 F1 1 − 2 2 2 a−b ( 2 )n ( )−1 2 cR aR + 1 2M λb e = 1− =1+ m R (bR2 + 1) pr (R) = 0. eνb /2 =

(47) (48) (49)

Using the boundary condition Eqs. (47-49), we arrived at [ √ ( )3n/2 ( 2 )1−m/2 2M A = 1− 1 − c3/2 F (R)R2 aR2 + 1 bR + 1 R { ( )3n/2 ( 2 )1−m/2 √ c3/2 R2 F (R) aR2 + 1 bR + 1 + 2b c(m − 2)R2 (

b b−a

)n/2

( B = Abc(m − 2)R

2

(

b b−a

)n/2

(

2

aR + 1

( 2 )1−m/2 √ bR + 1 + 2b c(m − 2)R2 (

2

)m (

)3n/2

(

R2

(50) [

b b−a

) −1

bR + 1 1 n 2 1 − 2M/R (aR + 1) ( ) m n m abR2 + a F (R) = 2 F1 1 − , − ; 2 − ; 2 2 2 a−b c=

}−1 ]−1

)n ( 2 )m/2 aR + 1 bR + 1 2

( )3n/2 c3/2 R2 F (R) aR2 + 1

)n/2

( 2 )n ( 2 )m/2 aR + 1 bR + 1

]−1

(51) (52) (53)

where the quantities M, R, a and b are chosen as free parameters to obtain wellbehaved in all respects. The parameters M and R are chosen from observed values of compact stars and a, b has to be tune so that the solution is well behaved in all respects. Now the gravitational red-shift is given by z(r) = e−ν/2 − 1

(54)

and it has decreasing in nature, Fig. 8 (Right). 7. Results and conclusion We have explored a new family of solutions satisfying Einstein’s field equations, Karmarkar condition in the background of Pandey-Sharma condition. Various analysis have been adopted to check the physical validity of the solutions and found that

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these solutions are suitable to explain the intrinsic characteristic of massive fluid spheres. Accordingly, we have constructed a model for Vela X-1 which is a X-ray pulsar. Table 1. The numerical values of central density, surface density, central pressure and surface red-shift are obtained for some well known compact star. Objects

a

b

c

n

m

A

B

R

M/M⊙

u

ρc × 1016 (gm/cc)

pc × 1035 (dyne/cm2 )

Vela X-1 LMC X-4 SMC X-4 Cen X-3

0.003 0.002 0.004 0.0035

0.0008 0.0006 0.00075 0.001

0.00649 0.00452 0.00458 0.00494

2 4 2 4

7 11 7 11

2.92 2.84 3.27 2.13

0.0318 0.030 0.031 0.031

9.56 8.3 8.83 9.178

1.77 1.04 1.29 1.49

0.370 0.251 0.292 0.325

2.63 1.83 1.85 1.99

13.3 7.06 10.8 13.3

The robustness of our solutions is that they satisfy all the energy conditions (Fig. 4 and 5 (Left)), stability factor within the limit and fulfill the stability condition. Since the pressure and density are monotonically deceasing in nature radially outward. The solution represents a static and equilibrium configuration as the gravitational force is counter-balanced by the combined force due to hydrostatic pressure and pressure anisotropy Fig. 7 (Right). The variation of mass at the interior and its compactness parameter with radial coordinate are displayed in Fig. 3 (Right). The mass and radius of Vela X-1 ((1.77 ± 0.08)M⊙ , 9.56 ± 0.08km) are matched quite well within the observed values by.38 Acknowledgments Farook Rahaman would like to thank the authorities of the Inter-University Centre for Astronomy and Astrophysics, Pune, India for providing research facilities. Farook Rahaman and Nayan Sarkar are also grateful to DST-SERB and CSIR (Grand No.-09/096(0863)/2016-EMR-I.) Govt. of India for financial support respectively. Appendix A. Deduction to equation (20) Equation (17) can be re-written as eλ 1 k2 = + . ν ′2 eν ν ′2 eν 4 Differentiating w.r.t. r and after simplification we get eλ ν ′ λ′ = −2(ν ′2 + ν ′′ ) + ν ′2 1 − eλ or ν ′ λ′ = −2(ν ′2 + ν ′′ ) + ν ′2 + λ′ ν ′ . 1 − eλ Also differentiating (17) w.r.t. r we can write λ′ =

k 2 ′ ν−λ ′′ νe (2ν + ν ′2 ). 4

(A.1)

(A.2)

(A.3)

(A.4)

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On using (A.2) we can re-express (A.4) as λ′ k 2 eν ν ′ λ′ . (A.5) = ′ ν 4 eλ − 1 Using (17), (A.3) and (A.5) the expression for anisotropy in (7) reduces to [ ][ ] 2 ν′ λ′ ν ′ eν − ∆= λ λ 1− 4e e − 1 r 2rB 2 or [ ][ ′ ν ] ν′ 2 λ′ νe ∆= λ − λ −1 4e r e − 1 2rB 2 which is the required equation. References 1. E. Witten, Phys. Rev. D 30 (1984) 272. 2. E. Farhi, R. Jaffe, Phys. Rev. D 30 (1984) 2379. 3. Ch. Kettner, F. Weber, M. K. Weigel, N. K. Glendenning, Phys. Rev. D 51 (1995) 1440. 4. M. Dey, I. Bombaci, J. Dey, S. Ray. B. C. Samanta, Phys. Lett. B 438 (1998) 123. 5. J. E. Horvath, J. A. D. F. Pacheco, Int. J. Mod. Phys. D 7 (1998) 19. 6. V. S. U. Maseswari, J. N. De, S, K, Samaddar, Phys. Rev. D 57 (1998) 3242. 7. J. B. Hartle, R. Sawyer, D. Scalapino, Astrophys. J. 199 (1995) 471. 8. M.K. Gokhroo, A.L. Mehra, Gen. Relativ. Gravit. 26 (1994) 75. 9. B. R. Iyer, C. V. Vishveshwara, General Relativistic Aspects of Neutron Star Models, in: A Random Walk in Relativity and Cosmology, Eds. N. Dadhich, J. Krishna Rao, J. V. Narlikar, and C. V. Vishveshwara (Wiley Eastern Limited, New Delhi, 1985), p. 109. 10. R. Tikekar, K. Jotania. Gravit. Cosmol. 15 (2009) 129. 11. Y.K. Gupta, M.P. Goel, Gen. Rel. Grav. 6 (1975) 499. 12. K. N. Singh, N. Pant, Eur. Phys. J. C 361 (2016) 177. 13. K. N. Singh, N. Pant, O. Troconis, Ann. Phys. 377 (2016) 256. 14. K. N. Singh, N. Pant, M. Govender, Eur. Phys. J. C 77 (2017) 100. 15. K. N. Singh, N. Pant, M. Govender, Chinese Phys. C 41 (2017) 015103. 16. K. N. Singh, M. H. Murad, N. Pant, Eur. Phys. J. A 53 (2017) 21. 17. P. Bhar, K. N. Singh, T. Manna, Int. J. Mod. Phys. D 26 (2017) 1750090. 18. P. Bhar, K. N. Singh, F. Rahaman, N. Pant, S. Banerjee, Int. J. Mod. Phys. D 26 (2017) 1750078. 19. P. Bhar, S. K. Maurya, Y. K. Gupta, T. Manna, Eur. Phys. J. A 52 (2016) 191 20. S. K. Maurya, M. Govender, Eur. Phys. J. C 77 (2017) 347. 21. S. K. Maurya, Y. K. Gupta, T. T. Smitha, F. Rahaman, Eur. Phys. J. A 52 (2016) 191. 22. S. K. Maurya, B. S. Ratanpal, M. Govender, Annals of Phys. 382 (2017) 36. 23. S. K. Maurya, M. Govender, Eur. Phys. J. C 77 (2017) 420. 24. E. Kasner, Am. J. Math. 43, (1921) 130. 25. L. Herrera, N. O. Santos, Phys. Rep. 286 (1997) 53. 26. K.R. Karmarkar, Proc. Ind. Acad. Sci. A 27 (1948) 56. 27. S. N. Pandey, S. P. Sharma, Gen. Relativ. Gravit. 14 (1981) 113. 28. S.K. Maurya, Y.K. Gupta, Saibal Ray, S. R. Chowdhury, arXiv:1506.02498 [gr-qc] (2015a)

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