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Relativistic Modelling of Stable Anisotropic Super-Dense Star
Vol. 76 (2015)
REPORTS ON MATHEMATICAL PHYSICS
No. 1
RELATIVISTIC MODELLING OF STABLE ANISOTROPIC SUPER-DENSE STAR S. K. M AURYA1 , Y. K. G UPTA2 and M. K. JASIM1 1 Department
of Mathematical & Physical Sciences, College of Arts & Sciences, University of Nizwa, Nizwa- Sultanate of Oman 2 Department of Mathematics, Jaypee Institute of Information Technology University, Sector-128 Noida (U.P.), India (e-mails: [email protected], [email protected]; [email protected]; [email protected]) (Received November 24, 2014 – Revised May 12, 2015) In the present article we have obtained new set of exact solutions of Einstein field equations for anisotropic fluid spheres by using the Herrera et al. [1] algorithm. The anisotropic fluid solutions so obtained join continuously to the Schwarzschild exterior solution across the pressurefree boundary. It is observed that most of the new anisotropic solutions are well-behaved and are used to construct the super-dense star models such as neutron stars and pulsars. Keywords: anisotropic fluids, anisotropic factor, Einstein’s equations, Schwarzschild solution, neutron stars, pulsars.
1.
Introduction
The first ever exact solution of Einstein’s field equations for a compact object in static equilibrium was obtained by Schwarzschild in 1916. The static isotropic and anisotropic exact solutions describing stellar-type configurations have continuously attracted the interest of physicists like Herrera et al. [1, 41]. Tolman [2] has proposed an easy way to solve Einstein’s field equations by introducing an additional equation necessary to give a determinate problem in the form of some ad hoc relation between the components of metric tensor. According to this methodology, he obtained eight solutions of the field equations, and his important approach still is used in obtaining the exact interior solutions of the gravitational field equations for fluid spheres. Buchdahl [3] proposed a famous bound on the mass radius ratio of relativistic fluid spheres as 2GM/c2 r ≤ 8/9, which is an important contribution in order to study stability of the fluid spheres. Also Ivanov [4] has given the upper bound of the red shift for realistic anisotropic star models which cannot exceed the values of 3.842, provided the tangential pressure satisfies a strong energy condition (ρ ≥ pr + 2pt ) and when the tangential pressure satisfies the dominated energy condition (ρ ≥ pt ). [21]
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S. K. MAURYA, Y. K. GUPTA and M. K. JASIM
Buchdahl [3] also obtained a nonsingular exact solution by choosing a particular choice of the mean density inside the star. The theoretical investigations of realistic fluid models indicate that stellar matter may be anisotropic at least in certain density ranges (ρ > 1015 gm/cm3 ) (Ruderman [5] and Canuto [6]) and radial pressure may not be equal to the tangential pressure of stellar structure. The existence of a solid core due to the presence of anisotropy in the pressure was thought of for type-3A super-fluid (Kippenhahm and Weigert [7]), different form of phase transitions (Sokolov [8]) or for other physical phenomena. On the scale of galaxies, Binney and Tremaine [9] have considered anisotropies in spherical galaxies, from a purely Newtonian point of view. The mixture of two gases (e.g. ionized hydrogen and electrons or monatomic hydrogen) can be described formally as an anisotropic fluid (Letelier [10] and Bayin [11]). The importance of equations of state for relativistic anisotropic fluid spheres have been investigated by generalizing the equation of hydrostatic equilibrium to include the effects of anisotropy (Bowers and Liang [12]). Their study shows that anisotropy may have nonnegligible effects on parameters such as maximum equilibrium mass and surface red shift. The relativistic anisotropic neutron star models with high densities by means of several simple assumptions showed that there is no limiting mass of neutron stars for arbitrary large anisotropy which have been studied by Heintzmann and Hillebrandt [13]. However, maximum mass of a neutron star still lies beyond 3–4 M⊙ . Also, the solutions for an anisotropic fluid sphere with uniform density and variable density have been studied by Maharaj and Maartens [14] and Gokhroo and Mehra [15], respectively. Most of the astronomical objects have variable density. Therefore, interior solutions of anisotropic fluid spheres with variable density are more realistic physically. Many workers have obtained different exact solutions for isotropic and anisotropic fluid spheres in different contexts (Delgaty and Lake [16], Dev and Gleiser [17], Komathiraj and Maharaj [18], Thirukkanesh and Ragel [19], Sunzu et al. [20], Harko and Mak [21], Mak and Harko [22], Chaisi and Maharaj [23], Maurya and Gupta [24–26], Feroze and Siddiqui [27], Pant et al. [28–30], Bhara et al. [31], Monowar et al. [32], Kalam et al. [33], Consenza et al. [34], Krori [35], Singh et al. [36], Patel and Mehta [37], Malaver [38, 39], Escupli et al. [40], Herrera and Santos [41, 42], Herrera et al. [43, 44]). The present paper consists of nine sections, Section 1 contains introduction; Section 2 contains metric, its components and the field equations. Section 3 embodies the solutions of anisotropic fluid spheres in different contexts. Section 4 contains the expressions for density and pressure which are mentioned for each fluid sphere. Section 5 describes various physical conditions to be satisfied by the anisotropic fluid spheres. The analytical behaviours of solutions under physical conditions (mentioned in Section 5) are mentioned in Section 6. Section 7 describes the evaluation of arbitrary constants involved in the fluid solutions by means of the smooth joining of Schwarzschild metric at the pressure-free interface r = a. The stability of models is proposed in Section 8, and finally Section 9 includes the physical analysis of the solutions so obtained along with the concluding remarks.
RELATIVISTIC MODELLING OF STABLE ANISOTROPIC SUPER-DENSE STAR
2.
23
Metric, components and field equations
The line element of a static spherical symmetric space-time in the curvature coordinates x i = (t, r, χ, ξ ) can be postulated as ds 2 = B 2 (r)dt 2 − ψ −1 (r)dr 2 − r 2 (dχ 2 + sin2 χ dξ 2 ),
(2.1)
while Einstein’s field equations are given as 1 j j j −κTi = Ri − Rδi , 2
(2.2)
where κ = 8πG/c4 . The components of the energy momentum tensor for spherically symmetric anisotropic fluid distribution are postulated in the form j
j
Ti = (c2 ρ + pt )vi v j − pt δi + (pr − pt )χi χ j ,
(2.3)
where v i is four-velocity B −1 v i = δ4i , χ i is the unit space-like vector in the direction √ of radial vector, χ i = ψδ1i , ρ is the energy density, pr is the pressure in direction of χ i (normal pressure) and pt is the pressure orthogonal to χi (transversal or tangential pressure). Suppose radial pressure is not equal to the tangential pressure, i.e. pr 6 = pt . If the radial pressure is equal to the transversal pressure, i.e. pr = pt , then it corresponds to isotropic or perfect fluid distribution. Let the measure of anisotropy be 1 = κ(pt − pr ), it is called the anisotropy factor (Herrera and Ponce de Leon [45]). The term 2(pt − pr )/r appears in the conservation equations Tji;i = 0 (where semi colon denotes the covariant derivative) which is representing a force due to anisotropic nature of the fluid. When pt > pr , then the direction of force is in the outward direction and inward when pt < pr . However, if pt > pr , then the force allows to construct a more compact object for anisotropic fluid than for isotropic fluid (Gokhroo and Mehra [15]). In view of the metric (2.1), the Einstein field equations (2.2) give 1−ψ ψ′ 8πG 2B ′ ψ ψ −1 8πG ρ = − , p = + , (2.4) r 2 2 4 c r r c Br r2 ′ ′′ 1 B B′ 1 1 ′ B ψ + + 2ψ − − 2 =2 1− 2 , (2.5) B r B rB r r where the “dash” denotes the derivative with respect to r. 3.
Classes of solutions
The field Eq. (2.5) has two dependent variables B(r) and ψ(r), therefore Eq. (2.5) can admit infinitely many solutions for different choices of B(r) and ψ(r) but all these solutions may or may not satisfy the physical conditions for the fluid spheres. For a given B(r), Eq. (2.5) reduces to a first-order ordinary differential equation in ψ(r). For its physically valid solution we will have to choose the metric potential
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S. K. MAURYA, Y. K. GUPTA and M. K. JASIM
B(r) such that B(0) is positive and finite. This is a sufficient condition for a static fluid sphere to be regular at the centre. Let us take B(r) of the form B(r) = D(1 − c0 r 2 )n , where n 6 = 0 and D is a positive arbitrary constant. (3.1)
Maurya and Gupta [22, 23] have already obtained all possible anisotropic solutions of Einstein field equations for n a positive integer ≥ 1 with c0 < 0, and for −1 < n < 0, c0 > 0, with different anisotropic factor 1. But in the present problem we consider B(r) = D(1 − c0 r 2 )n with n = −1, −2 and −3 and the anisotropy factor
10 c0 (c0 r 2 )−n ; (3.2) [1 − (n + 1)c0 r 2 ]1+n where c0 and 10 are positive constants. Herrera et al. [1] have proposed an algorithm for all possible spherically symmetric anisotropic solutions of Einstein field equations. By using the Herrera et al. [1] algorithm, Eq. (2.5) reduces to ′ 2y 6 4 2 1 ψ′ + + 2y − + 2 ψ= 1− 2 , (3.3a) y r r y y r where B ′ (r) 1 y(r) = + . (3.3b) 2B(r) r Integrating (3.3a) we can obtain ψ as " Z # R 2 y(r)(1 + 1(r)r 2 )e [(4/r y(r))+2y(r)]dr 6 r −2 dr + A r8 R ψ= , (3.3c) 2 y 2 (r)e [(4/r y(r))+2y(r)]dr and [1 − (n + 1)c0 r 2 ] y(r) = . (3.3d) r(1 − c0 r 2 ) Further Z −2φ Z −2φ e e 3 2φ ψ = φf e 10 dφ − dφ + Aφf 3 e2φ for n = −1, (3.4a) 2 f φf 2 and Z Z φf 2−n f n−1 f n−1 φf 2−n ψ = 2/(n+1) 10 dφ − dφ + A ; g φ n+1 g n(n+3)/(n+1) φ 2 g (n−1)/(n+1) g 2/(n+1) for n 6= −1, (3.4b) 1=
where φ = c0 r 2 , f = (1 − φ) and g = [1 − (n + 1)φ].
RELATIVISTIC MODELLING OF STABLE ANISOTROPIC SUPER-DENSE STAR
25
Eqs. (3.4a) and (3.4b) give the following solutions: ψ = Aφf 3 e2φ + (1 − 2φ)f 2 − (6 + 10 )f 3 φe2φ−2 Ei(2 − 2φ) + 10 φf 2 ; for n = −1, (3.5a) where Ei(2 − 2φ) = log(2 − 2φ) +
∞ P
N =1
(2−2φ)N N!N
,
4 2 f 15 2 4 2 (15φ − 25φ + 8) 2 4 ψ = Aφ(1 + φ) f + f + φ(1 + φ) f log 8 16 1+φ 10 1−φ + φf 2 2(φ 2 − φ + 2) + f 2 (1 + φ)2 log ; for n = −2, 16 1+φ (3.5b) 1 5 3h 1 + 2φ ψ = Aφf 5 (1 + 2φ) − f − 320φ log (1 + 2φ) 3 243 f f + 10 φf 3 (1 + 2φ)(1 − 3φ + 3φ 2 ) ; for n = −3, (3.5c) where h = (−81 + 130φ + 36φ 2 − 720φ 3 + 320φ 4 ). 4.
Expressions for energy density, pressures for different values of n
(a) For n = −1,
8πG ρ c2
8πG pr c4
8πG pt c4
= c0 [Af 2 e2φ (4φ 2 + 5φ − 1) + (6 + 10 )f 2 (3 − 5φ − 4φ 2 )e2φ−2 Ei(2 − 2φ) + (6 − 11φ 2 + 2φ 3 ) + 10 [(−3 + 10φ − 7φ 2 ) ¯ − 2φ)]], + 2φe2φ−2 f 3 Ei(2
(4.1a)
= c0 [Af 2 (1 + φ)e2φ + (2 − 10φ + 7φ 2 − 2φ 3 ) − 6f 2 (1 + 5φ)e2φ−2 Ei(2 − 2φ)
+10 (1 − φ 2 )[1 − e2φ−2 f Ei(2 − 2φ)]],
(4.2a)
= c0 [Af 2 (1 + φ)e2φ + (2 − 10φ + 7φ 2 − 2φ 3 ) − 6f 2 (1 + 5φ)e2φ−2 Ei(2 − 2φ)
+10 [(1 + φ − φ 2 ) − (1 − φ 2 )e2φ−2 f Ei(2 − 2φ)]]. ¯ where Ei(2 − 2φ) =
−2 (2−2φ)
−2
∞ P
N=1
(2−2φ)N−1 . N!
(4.3a)
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S. K. MAURYA, Y. K. GUPTA and M. K. JASIM
(b) For n = −2,
8πG ρ c2
48+85φ−350φ 2 +90φ 3 +330φ 4 −195φ 5 +30φf 3 (1+φ) 8 1 3 1−φ 2 3 3 2 − f 45−15φ−285φ −225φ −10 (15φ +19φ +φ−3) log 16 1+φ 10 f [11φ 3 −16φ 2 +19φ−6+2f 2 φ(1+φ)] − 8
= c0
+Af 3 (−3+φ+19φ 2 +15φ 3 ) , (4.1b) 8πG 1 2 3 2 3 4 5 p = c A(1+3φ)f (1+φ) + (16−49φ−50φ +90φ +30φ −45φ ) r 0 c4 8 1−φ (15+10 ) 3 2 f (1+3φ)(1+φ) log + 16 1+φ 10 (4.2b) + f (1+3φ)(4−2φ+2φ 2 ) , 16 8πG (16−49φ−50φ 2 +90φ 3 +30φ 4 −45φ 5 ) 3 2 = c A(1+3φ)f (1+φ) + p 0 t c4 8 (15+10 ) 3 1−φ + f (1+3φ)(1+φ)2 log 16 1+φ 10 (4.3b) + (4+6φ+2φ 2 +26φ 3 −6φ 4 ) . 16 (c) For n = −3,
8πG ρ c2
= c0 −A(3−15φ+90φ 3 −165φ 4 +117φ 5 −34φ 6 ) 1 (21320−26565φ−55034φ 2 +156004φ 3 −11040φ 4 +29120φ 5 ) 81 320 4 1+2φ + f (−1+5φ+26φ 2 ) log 243 1−φ 2 2 3 4 −10 f (3−14φ−10φ +93φ −90φ ) , (4.1c) +
8πG pr c4
= c0 Af 4 (1+7φ+10φ 2 )−(−356+411φ+146φ 2−3820φ 3−1680φ 4 +1600φ 5 ) 320 4 1+2φ + f (1+2φ)(1+5φ) log 243 1−φ 2 2 3 4 +10 f (1+4φ−8φ −9φ +30φ ) , (4.2c)
27 = c0 Af 4 (1+7φ+10φ 2 )+(356−411φ−146φ 2 +3820φ 3 +1680φ 4−1600φ 5 ) 320 4 1+2φ + f (1+2φ)(1+5φ) log 243 1−φ 2 2 3 4 3 2 +10 f (1+4φ−8φ −9φ +30φ )+φ (1+2φ) . (4.3c)
RELATIVISTIC MODELLING OF STABLE ANISOTROPIC SUPER-DENSE STAR
5.
8π G pt c4
Reality and physical (well-behaved) conditions for anisotropic solutions
The physically meaningful anisotropic solution for the Einstein’s field equations must satisfy some physical conditions (Mak and Harko [20], Maurya and Gupta [22]) such as: (i) The solution should be free from physical and geometrical singularities, i.e. the pressure and energy density should be finite at the centre and the metric potentials B(r) and ψ(r) must have nonzero positive values. (ii) The radial pressure pr must be vanishing but the tangential pressure pt may not vanish at the boundary r = ra of the fluid sphere. However, the radial pressure is equal to the tangential pressure at the centre of the fluid sphere. (iii) The density ρ, radial pressure pr and tangential pressure pt should be positive inside the star. (iv) (dpr /dr)r=0 = 0 and (d 2 pr /dr 2 )r=0 < 0, so that the pressure gradient dpr /dr is negative for 0 ≤ r ≤ ra . (v) (dpt /dr)r=0 = 0 and (d 2 pt /dr 2 )r=0 < 0, so that the pressure gradient dpt /dr is negative for 0 ≤ r ≤ ra . (vi) (dρ/dr)r=0 = 0 and (d 2 ρ/dr 2 )r=0 < 0, so that the density gradient dρ/dr is negative for 0 ≤ r ≤ ra . The conditions (iv)–(vi) imply that the pressure and density should be maximal at the centre and monotonically decreasing towards the surface. (vii) Inside the fluid ball the speed of sound should be less than that of light, i.e. s s dpr dpt < 1, 0≤ < 1. 0≤ 2 c dρ c2 dρ In addition, the velocity of sound should be monotonically decreasing away from the centre, i.e. 2 2 d pr d dpt d pt d dpr < 0 or > 0 and < 0 or >0 2 dr dρ dρ dr dρ dρ 2
for 0 ≤ r ≤ ra . In this context, it is worth mentioning that the equation of state at ultra-high distribution has the property that the speed of sound is decreasing outwards (Canuto [30]). (viii) A physically reasonable energy-momentum tensor has to obey the energy conditions ρ ≥ pr + 2pt (strong energy condition) and ρ + pr + 2pt ≥ 0.
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S. K. MAURYA, Y. K. GUPTA and M. K. JASIM
(ix) The red shift at center Z0 and at surface Za should be positive, finite and both bounded. (x) The anisotropy factor 1 should be zero at the center and must be increasing towards the surface. 6.
Physical properties of the new solutions (a) For n = −1 The expressions for pressure and density at the centre are given as: 10 e2 − Ei(2)(6 + 10 ) 8πG 8πG = = c0 A + 2 + , pr pt c4 c4 e2 r=0 r=0 8πG Ei(2)(18 + 310 ) − 3e2 10 ρ = c0 −A + 6 + . c2 e2 r=0
(6.1a) (6.2a)
The pressure and density should be positive at the centre and consequently A satisfies the following inequalities Ei(2)(6 + 10 ) − 10 e2 Ei(2)(18 + 310 ) − 310 e2 and A < 6 + . A > −2 + e2 e2 (6.3a) The ratio of pressure-density should be positive and less than 1 at the centre, i.e. p0 /ρ0 c2 ≤ 1 which gives the following inequality, pr pt e2 (A + 2) + 10 e2 − Ei(2)(6 + 10 ) = = ≤ 1. (6.4a) ρc2 r=0 ρc2 r=0 e2 (−A + 6) + Ei(2)(18 + 310 ) − 310 e2
From Eq. (6.4a) we get
Ei(2)(6 + 10 ) − 210 e2 . e2 Using Eqs. (6.3a) and (6.5a), A satisfies the following inequalities Ei(2)(6 + 10 ) − 210 e2 Ei(2)(6 + 10 ) − 10 e2
(6.5a)
10 ≥ 0. (6.6a)
Differentiating (4.2a) with respect to r, we get an expression for the gradient of pressure h 8πG dpr 2 r Ae2φ (1 − 4φ + φ 2 + 2φ 3 ) + (−10 + 14φ − 6φ 2 ) = 2c 0 c4 dr ¯ − 6f 2 (1 + 5φ)e2φ−2 Ei(2 − 2φ) − 6(5 − 12φ − 3φ 2 + 10φ 3 )e2φ−2 Ei(2 − 2φ)
+ 10 [−2φ + e2φ−2 Ei(2 − 2φ)(−2 + 5φ − 3φ 3 ) i ¯ − e2φ−2 f 2 (1 + φ)Ei(2 − 2φ)] .
(6.7a)
RELATIVISTIC MODELLING OF STABLE ANISOTROPIC SUPER-DENSE STAR
29
Thus it is found that the extremum of pr occurs at the centre, i.e. and
pr′ = 0 ⇒ r = 0 2c02 8πG ′′ ¯ ¯ p = − (10 − A)e2 + 6Ei(2) + 30Ei(2) + 10 [2Ei(2) + Ei(2)] . r r=0 4 2 c e (6.8a)
This shows that the expression on the right-hand side of Eq. (6.8a) is negative for all values of A and 10 satisfying condition (6.8a). Then the radial pressure (pr ) has maximum at the centre and is monotonically decreasing. Differentiating (4.3a) with respect to r we get h 8π G dpt 2 2φ 2 3 2 = 2c r 0 Ae (1 − 4φ + φ + 2φ ) + (−10 + 14φ − 6φ ) + 10 (1 − 2φ) c4 dr + e2φ−2 Ei(2 − 2φ)[10 (−2 + 5φ − 3φ 3 ) − 6(5 − 12φ − 3φ 2 + 10φ 3 )] i ¯ − (6 + 10 )f 2 (1 + 5φ)e2φ−2 Ei(2 − 2φ) , (6.9a)
this suggests that the extremum value of pt occurs at the centre, i.e.
pt′ = 0 ⇒ r = 0 (6.10a) and 2c02 8π G ′′ 2 ¯ ¯ p = − (10 − A − 1 )e + 6 Ei(2) + 30Ei(2) + 1 [2Ei(2) + Ei(2)] . 0 0 t r=0 c4 e2 (6.11a) Under the condition (6.6a), the expression on the right-hand side of Eq. (6.11a) is negative for all values of A and 10 . This shows that the tangential pressure (pt ) is maximum at the centre and it is monotonically decreasing. Now, differentiating Eq. (4.1a) with respect to r, we get h 8πG dρ 2 2φ 2 3 4 = 2c r 0 Ae (5 − 23φ + 10φ + 8φ ) c2 dr + (−30 − 24φ − 60φ 3 − 48φ 4 )e2φ−2 Ei(2 − 2φ) ¯ + 6f 2 (3 − 5φ − 4φ 2 )e2φ−2 Ei(2 − 2φ) − 22φ + 6φ 2
where
+ 10 [e2φ−2 f {Ei(2 − 2φ)(−5 − 19φ + 12φ 2 + 8φ 3 ) ¯ − f (−5 + 9φ + 8φ 2 )Ei(2 − 2φ) i ¯¯ − 2f 2 φ Ei(2 − 2φ)} − 2(−5 + 7φ)] , (6.12a)
¯¯ Ei(2 − 2φ) =
∞ X N (2 − 2φ)N −1 4 + 4 . (2 − 2φ)2 (N + 1)! N=1
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S. K. MAURYA, Y. K. GUPTA and M. K. JASIM
The extremum of ρ occurs at the centre, i.e. ρ′ = 0 ⇒ r = 0
and
2c02 8πG ′′ ¯ ρ = − −(5A + 1010 )e2 + Ei(2)(30 + 510 ) − Ei(2)(18 + 510 ) . r=0 2 2 c e (6.13a)
Thus, the expression on the right-hand side of (6.12a) is negative. This shows that the density ρ has maximum at the centre and is monotonically decreasing towards the pressure-free interface. 1 dp The square of adiabatic sound speed at the centre, 2 , is given by c dρ r=0 ¯ ¯ 1 dpr (10 − A)e2 + 6Ei(2) + 30Ei(2) + 10 [2Ei(2) + Ei(2)] = , (6.14a) ¯ c2 dρ r=0 (5A + 1010 )e2 + Ei(2)(30 + 510 ) + Ei(2)(18 + 510 ) ¯ ¯ 1 dpt (10 − A − 10 )e2 + 6Ei(2) + 30Ei(2) + 10 [2Ei(2) + Ei(2)] . = ¯ c2 dρ r=0 (5A + 1010 )e2 + Ei(2)(30 + 510 ) + Ei(2)(−18 + 510 ) (6.15a) The causality condition is obeyed at the centre for all values of constants under the condition (6.6a), i.e. the radial and tangential velocities of sound are monotonically decreasing and less than 1. (b) For n = −2
The central values of pressure and density are given by 8πGpt c0 8πGpr = = (4A + 8 + 10 ) , 4 4 c c 4 r=0 r=0 8πGρ c0 = (−12A + 24 + 310 ) . 2 c 4 r=0
(6.1b) (6.2b)
The central values of pressure and density should be positive and finite, then A satisfies the following conditions: 10 10 A > −2 − and A < 2 − . (6.3b) 4 4 The condition that the ratio of pressure and density should be positive and less than 1 at the centre, i.e. ρpc02 ≤ 1 leads to the inequality 0 pr pt 4A + 8 + 10 = = ≤ 1. (6.4b) ρc2 r=0 ρc2 r=0 −12A + 24 + 310 From inequality (6.4b) we get
10 A≤ 1+ . 8
(6.5b)
RELATIVISTIC MODELLING OF STABLE ANISOTROPIC SUPER-DENSE STAR
31
Owing to inequalities (6.3b) and (6.5b), A satisfies the inequality 10 10
64 + 130x − 330x 2 − 150x 3 + 270x 4 8 10 + (2 − 16φ + 19φ 2 −10φ 3 −3φ 4 ) . (6.7b) 8 Thus, it is found that the extremal value of pr occurs at the centre, i.e. 8πG ′′ −16A − 64 + 210 ′ pr = 0 ⇒ r = 0 and pr r=0 = . (6.8b) c4 8 Hence, the expression on the right-hand side of Eq. (6.8b) is negative for all values of constants A and 10 satisfying condition (6.6b) and it shows that the pressure pr has maximum at the centre and is monotonically decreasing. Differentiating Eq. (4.3b) with respect to r we get 1 − φ 8πG dpt 15 − 10 2 f 2 (−1 + 3φ + 13φ 2 + 9φ 3 ) = 2c0 r −2A + log c4 dr 8 1 + φ −
64 + 130x − 330x 2 − 150x 3 + 270x 4 8 10 2 3 4 (6.9b) + (2 + 43φ −10φ −3φ ) , 8 Thus, it is found that the extremal value of pt occurs at the centre, i.e. 8πG ′′ 2 16A − 64 + 210 ′ pt = 0 ⇒ r = 0 and pt r=0 = 2c0 . (6.10b) c4 8 The expression on the right-hand side of Eq. (6.10b) is also negative for all values of A and 10 satisfying condition (6.6b). This behaviour shows that the transversal pressure pt has maximum at the centre and is monotonically decreasing. Now, differentiating Eq. (4.1b) with respect to r, the expression for the gradient of density is 8π G dρ 5S 15+10 1−φ 2 = 2c r − 2A+ log f 2 (−5−17φ +25φ 2 +45φ 3 ) 0 c2 dr 8(1−φ 2 ) 8 1+φ 10 2 3 3 − (−30+88φ −72φ +2φ +25φ ) , (6.11b) 8 −
where S = (32 + 86φ + 10φ 2 − 176φ 3 − 312φ 4 + 90φ 5 + 273φ 6 ).
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S. K. MAURYA, Y. K. GUPTA and M. K. JASIM
Thus, the extremal value of ρ occurs at the centre, i.e. 8πG ′′ ′ 2 180 + 80A + 3010 ρ r=0 = 2c0 ρ = 0 ⇒ r = 0 and > 0. c2 8
(6.12b)
The right-hand side of Eq. (6.12b) is positive due to the inequality (6.6b), and this condition gives that the energy density ρ is minimal at the centre and is monotonically increasing. 1 dp Also, the square of its adiabatic sound speed at the centre, 2 , is given c dρ r=0 by 1 dpt 1 dpr (16A − 64 + 310 ) = 2 = . (6.13b) 2 c dρ r=0 c dρ r=0 [180 + 80A + 3010 ]
The causality condition is negative at the centre for all values of constants satisfying condition (6.6b). Due to the increasing nature of energy density, the solution behaves not well for n = −2. Case 3: n = −3.
The central values of pressure and density are given by 8πGpt 8πGpr = = c0 [A + 356 + 10 ] , c4 c4 r=0 r=0 8πGρ c0 = [−243A + 21320 − 24310 ] . c2 81 r=0
(6.1c) (6.2c)
The central values of pressure and density should be positive and finite. Then A satisfies the following conditions: 21320 A > (−356 − 10 ) and A < − 10 . (6.3c) 243 The condition that the ratio of pressure-density is positive and less than 1 at the centre, i.e. ρpc02 ≤ 1, leads to the inequality 0 pr pt 81 [A + 356 + 10 ] = = ≤ 1. (6.4c) ρc2 r=0 ρc2 r=0 −243A + 21320 − 24310 Eq. (6.4c) leads to
7516 A≤− + 10 . 324
Using Eqs. (6.3c) and (6.5c) we get the inequalities for A as 7516 − (356 + 10 ) < A ≤ − + 10 . 324
(6.5c)
(6.6c)
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RELATIVISTIC MODELLING OF STABLE ANISOTROPIC SUPER-DENSE STAR
Differentiating (4.2c) with respect to r we get 8πG dpr 2 = 2c0 r Ap1 (φ) − (411 − 292φ − 11460φ 2 − 6720φ 3 + 8000φ 4 ) c2 dr 320 3 320 1 + 2φ + f (1 + 5φ) + p2 (φ) log 243 243 1−φ + 10 (2 − 30φ + 33φ 2 + 160φ 3 −345x 4 + 180φ 5 ) ,
where
(6.7c)
p1 (φ) = (3 − 24φ−6φ 2 + 132φ 3 −165φ 4 + 60φ 5 ),
p2 (φ) = (15 − 40φ − 252φ 2 − 336φ 3 + 120φ 4 ).
Thus, it is found that the extremal value of pr occurs at the centre, i.e. 8πG ′′ 320 2 p = 2c + 21 (6.8c) pr′ = 0 ⇒ r = 0 and 3A − 411 + 0 . r r=0 0 c4 243 So, the expression on the right-hand side of Eq. (6.8c) is negative for all values of A and 10 satisfying condition (6.6c). This shows that the pressure (pr ) has maximum at the centre and is monotonically decreasing. Differentiating (4.3c) with respect to r we get 8π G dpt 2 = 2c r A(3 − 24φ−6φ 2 + 132φ 3 −165φ 4 + 60φ 5 ) 0 c2 dr 320 3 − (411 − 292φ − 11460φ 2 − 6720φ 3 + 8000φ 4 ) + f (1 + 5φ) 243 320 1 + 2φ + (15 − 40φ − 252φ 2 − 336φ 3 + 120φ 4 ) log 243 1−φ (6.9c) + 10 (2 − 30φ + 36φ 2 + 176φ 3 −325x 4 + 180φ 5 ) . Thus, it is found that the extremum of pt occurs at the centre, i.e. 320 8πG ′′ 2 ′ + 210 . pt = 0 ⇒ r = 0 and 4 pt r=0 = 2c0 3A − 411 + c 243
(6.10c)
It is clear that the expression on the right-hand side of Eq. (6.10c) is negative for all values of A and 10 satisfying condition (6.6c). This condition gives that the transversal pressure is maximal at the centre and is monotonically decreasing. Now, differentiating equation (4.1c) with respect to r we get ρ1 8πG dρ 2 = 2c0 r A(15 − 270x 2 + 660x 3 − 585x 4 + 204x 5 ) + 2 c dφ 81 320 1 + 2φ 3 4 5 + (9 − 140x + 540x − 495x + 156x ) log 81 1−φ + 10 (2 − 30φ + 33x 2 + 160φ 3 −345x 4 + 180φ 5 ) , (6.11c)
34
S. K. MAURYA, Y. K. GUPTA and M. K. JASIM
ρ1 = (−26885−55018φ +64686φ 2 −1149772φ 3 +685612φ 4 −296960φ 5 +128960φ 6 ). Thus, the extremum of ρ occurs at the centre, i.e. 8πG ′′ 2 −26885 ρ ′ = 0 ⇒ r = 0 and ρ = 2c + 15A + 21 0 , 0 r=0 c2 81
(6.12c)
Hence, the expression on the right-hand side of Eq. (6.12c) is negative for all values of A and 10 satisfying condition (6.6c). Then the density ρ has maximum at the centre and is monotonically decreasing. , is given by The square of the adiabatic sound speed at the centre, c12 dp dρ r=0 1 dpt −99553 − 729A + 48610 1 dpr = = . (6.13c) c2 dρ r=0 c2 dρ r=0 3[−26885 + 1215A + 16210 ]
The causality condition is less than 1 and positive at the centre for all values of A and 10 satisfying condition (6.6c). 7.
Boundary conditions for evaluation of constants A, D and c0 The above system of equations is to be solved subject to the boundary condition that the radial pressure pr = 0 at r = a (where r = a is the outer boundary of the fluid sphere). It is clear that m(r = a) = M is a constant and, in fact, the interior metric (2.1) can be joined smoothly at the surface of sphere (r = a) to an exterior Schwarzschild metric whose mass is the same as above, i.e. m(r = a) = M (Masiner and Sharp [45]). Then the interior metric of this fluid sphere should be joined smoothly with Schwarzschild exterior metric such as B 2 (a) = 1 − 2u, where u = M/a, where M is the mass of the fluid sphere as measured by the exterior field and a is the boundary of the sphere. By joining (3.5a) and (3.5c) on the boundary of the anisotropic fluid sphere (r = a) and by setting φa = co a 2 and ψanis (a) = 1 − 2uanis , we get the expressions for mass for n = −1 and −3 as: a Manis = φa − Afa3 e2φa + (4φa − 3φa2 + 2φa3 ) 2 Manis =
+ (6 + 10 φa )fa3 e2φa −2 Ei(2 − 2φa ) − 10 φa fa2 ,
a M1 φa − A(1 − φa )5 (1 + 2φa ) 2 81 320 1 + 2φa 3 5 + (1 − φa ) (1 + 2φa ) log − M2 10 φa .fa , 243 1 − φa
M1 = (292 − 305φa − 662φa2 + 1796φa3 − 1360φa4 + 320φa5 ), M2 = (1 + 2φa )(1 − 3φa + 3φa2 ),
S = (16 + 17φa − 50φa2 + 10φa3 + 30φa4 − 15φa5 ).
(7.1)
(7.2)
35
RELATIVISTIC MODELLING OF STABLE ANISOTROPIC SUPER-DENSE STAR
The arbitrary constant A in (3.5a) and (3.5c) can be determined by putting the radial pressure pr equal to zero at the boundary, the expressions of constant A for n = −1 and −3 are given as: [6(1 + 5φa ) + 10 fa ]fa2 e2φa −2 Ei(2 − 2φa ) − (2 − 10φ + 7φ 2 − 2φ 3 ) − 10 fa2 ; (7.3) fa2 (1 + φa )e2φa (−356 + 411φa + 146φa2 − 3820φa3 − 1680φa4 + 1600φa5 ) 320 4 1 + 2φa 2 2 3 4 − − 10 fa (1 + 4φ − 8φ − 9φ + 30φ ) fa (1 + 2φa )(1 + 5φa ) log 243 1 − φa A= . fa4 (1 + 7φa + 10φa2 ) (7.4) A=
Also, the arbitrary constants D in the metric potential for the cases n = −1 and −3 can be computed by the condition B(a) = 1 − 2uanis , D = fa3 [Aφa fa e2φa + (1 − 2φa ) − (6 + 10 )fa φa e2φa −2 Ei(2 − 2φa ) + 10 φa ]; 1 + 2φa 1 2 3g (1 + 2φ ) fa D = fa6 Aφa fa2 (1 + 2φa ) − − 320φ log a a 243 fa3 fa +10 φa (1 + 2φa )(1 − 3φa + 3φa2 ) .
(7.5)
(7.6)
The positive constant c0 can be calculated 2 × 1014 by taking the surface density gm/cm3 and using the condition c0 = 8πc2G a 2 ρa / 1 − ψa − aψa′ for n = −1 and −3. Tables for numerical values of physical quantities
In Tables 1–2: Z = red shift, solar mass M⊙ = 1.475 km, G = 6.673 × 10 cm3 /gs2 , c = 2.997 × 1010 cm/s, D = (8π G/c2 c0 )ρ, Pr = (8π G/c4 c0 )pr a 2 , Pt = (8πG/c4 c0 )pt a 2 . −8
Table 1. n = −1, 10 = 4.2395, c0 a 2 = 0.0829, radius (a) = 16.0780 km, mass (M) = 1.7609 M⊙ . p p r/a Pr Pt D 1 dpr /c2 dρ dpt /c2 dρ pr /c2 ρ pt /c2 ρ Z 0.0
1.3165
1.3165
20.7045
0.0000
0.9999
0.9394
0.0636
0.0636
0.2570
0.2
1.2543
1.2683
20.5763
0.0140
0.9918
0.9312
0.0610
0.0616
0.2554
0.4
1.0723
1.1285
20.1933
0.0562
0.9680
0.9070
0.0531
0.0559
0.2504
0.6
0.7852
0.9117
19.5603
0.1265
0.9292
0.8672
0.0401
0.0466
0.2421
0.8
0.4167
0.6417
18.6860
0.2249
0.8761
0.8125
0.0223
0.0343
0.2303
1.0
0.0000
0.3515
17.5839
0.3515
0.8092
0.7427
0.0000
0.0200
0.2151
36
S. K. MAURYA, Y. K. GUPTA and M. K. JASIM
Table 2. n = −3, 10 = 0.7094, c0 a 2 = 0.0028, radius (a) = 3.1274 km, mass (M) = 0.8672 M⊙ . p p r/a Pr Pt D 1 × 1013 dpr /c2 dρ dpt /c2 dρ pr /c2 ρ pt /c2 ρ Z 0.0 4.0619
4.0619
13.1901
0
0.5160
0.5160
0.3079
0.3079
0.8597
0.2 3.8982
3.8982
13.1838
0.00997
0.5158
0.5158
0.2957
0.2957
0.8595
0.4 3.4077
3.4077
13.1650
0.63900
0.5153
0.5153
0.2588
0.2588
0.8590
0.6 2.5922
2.5922
13.1336
7.2949
0.5144
0.5144
0.1974
0.1974
0.8582
0.8 1.4549
1.4549
13.0896
41.116
0.5132
0.5132
0.1111
0.1111
0.8571
1.0 0.0000 1.5748*10−8 13.0331
157.48
0.5116
0.5116
0.0000
1.208*10−9 0.8556
8.
Stability of the stellar structure For physically acceptable model, one aspect is that the velocity of sound should be within the range 0 ≤ va2 = (dp/c2 dρ) ≤ 1 (Herrera [47] and Abreu et al. [48]). In the present models, the expression for velocity of sound at the centre is given by equations (6.13a), (6.14a) and (6.13c). We plot the radial and transversal velocities of sound in Fig. 3 and conclude that all parameters satisfy the inequalities 2 2 = (dpr /c2 dρ) ≤ 1 and 0 = vat = (dpt /c2 dρ) ≤ 1, everywhere inside the star. 0 = var 2 2 From Eqs. (6.13a), (6.14a) and (6.13c), we found that |vat − var | ≤ 1 at the centre and proved that the velocity of sound is monotonically decreasing inside the star. 2 2 2 2 Also 0 ≤ vat ≤ 1 and 0 ≤ var ≤ 1, therefore vat − var ≤ 1. Now, to examine the stability of local anisotropic fluid distribution we use Herrera’s [46] concept. Herrera [46] proposed the cracking (also known as overturning) concept which the states that the region, in which radial speed of the sound is greater than the transversal speed is a potentially stable region. In our proposed models, the models are stable with the radius 16.0780 km, 10 = 4.2395, c0 a 2 = 0.0829 for n = −1 and radius 3.1274 km, 10 = 0.7094, c0 a 2 = 0.0028 for n = −3. 9.
Physical analysis and conclusions In the present paper, we have presented the new set of anisotropic exact solutions of Einstein’s field equations by taking the metric potential g44 = D(1 − c0 r 2 )n for n = −1, −2 and −3 and the specific choice of anisotropic factor 1 which involves the anisotropic parameter 10 . The solutions so obtained are used to construct the super dense star models with the surface density 2 × 1014 gm/cm3 . It is observed that the solutions satisfy all reality and physical conditions (mentioned in Section 5) for n = −1 and −3. But the solution is not compatible for n = −2 due to the increasing nature of its density (Section 6, Case (b)). The anisotropic fluid sphere possesses the maximum mass and corresponding radii are 1.7609 M⊙ and 16.0780 km for n = −1, c0 a 2 = 0.0829, 10 = 4.2395 and 0.8672 M⊙ and 3.1274 km for n = −3, c0 a 2 = 0.0028, 10 = 0.7094. The red shift for n = −1 and −3 are monotonically decreasing towards the pressure free interface r = a and we have found that the
RELATIVISTIC MODELLING OF STABLE ANISOTROPIC SUPER-DENSE STAR
37
18 16 14 12 10 8
n = −1 PrPrforforn=-1 D − Pr − D-Pr-2Pt for2P n=-1 t for n = −1
PrPrforforn=-3 n = −3 D − Pr − D-Pr-2Pt for2P n=-3 t for n = −3
6 4 2 0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
r/a Fig. 1. Variations of radial pressure Pr and the trace D − Pr − 2Pt of the energy-momentum tensor for n = −1, c0 a 2 = 0.0829, 10 = 4.2395 and n = −3, c0 a 2 = 0.0028, 10 = 0.7094.
1 0.9
Vr Vr for forn=-1 n = −1
0.8
R forn=-1 n = −1 Rrr for
0.7 0.6 0.5
Vr for forn=-3 n = −3 Vr
Rrr for R forn=-3 n = −3
0.4 0.3 0.2
0.1 0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
1
r/a p Fig. 2. Variations of the radial velocity Vr = dpr /c2 dρ and ratio of radial pressure and density Rr = pr /c2 ρ of the energy-momentum tensor for n = −1, c0 a 2 = 0.0829, 10 = 4.2395 and n = −3, c0 a 2 = 0.0028, 10 = 0.7094.
red shift at the centre (Z0 ) and at the surface (Za ) are: (i) Z0 = 0.2570 and Za = 0.2151 for n = −1, (ii) Z0 = 0.8597 and Za = 0.8556 for n = −3 for both strong energy and dominated energy conditions. In our models, the red shift also satisfied the upper bound limit for the realistic anisotropic star models (Ivanov [4])
38
S. K. MAURYA, Y. K. GUPTA and M. K. JASIM
1 0.9 0.8
Vt Vt for forn=-1 n = −1
0.7
Rt for forn=-1 n = −1 Rt
0.6
Vt Vt for forn=-3 n = −3
0.5
Rt Rt for forn=-3 n = −3
0.4 0.3 0.2 0.1 0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
1
r/a p Fig. 3. Variations of the tangential velocity Vt = dpt /c2 dρ and the ratio of tangential pressure and density Rt = pt /c2 ρ of the energy-momentum tensor for n = −1, c0 a 2 = 0.0829, 10 = 4.2395 and n = −3, c0 a 2 = 0.0028, 10 = 0.7094. 42
anisotropic anisotropy factor (delta1) (11 ) forfornn=-1 = −1
36
anisotropyfactor factor(delta2) (12 ) for = −3 ansotropy for n n=-3
30 24 18
12 6 0 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
1
r/a Fig. 4. Variations of the anisotropy factor 11 = 1 × 10−2 and anisotropy factor 12 = 1 × 10−9 of the energy-momentum tensor for n = −1, c0 a 2 = 0.0829, 10 = 4.2395 and n = −3, c0 a 2 = 0.0028, 10 = 0.7094.
and its behaviour is represented by Fig. 5. The Tables 1 and 2 show the values of physical parameters. Fig. 1 shows that the fluid spheres satisfies energy condition. The behaviours of velocity and pressure density ratios by Figs. 2 and 3. Fig. 4 represents the increasing nature of anisotropy the fluids spheres.
numerical the strong are given factor for
RELATIVISTIC MODELLING OF STABLE ANISOTROPIC SUPER-DENSE STAR
39
1 0.9 0.8 0.7 0.6 0.5
0.4 0.3 0.2
red shift shiftfor forn=-1 n = −1 red
0.1
red shift shiftfor forn=-3 n = −3 red
0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
1
r/a Fig. 5. Variations of the red-shift (Z) of the energy-momentum tensor for n = −1, c0 a 2 = 0.0829, 10 = 4.2395 and n = −3, c0 a 2 = 0.0028, 10 = 0.7094.
Acknowledgements The authors are very grateful to the Referees for their valuable comments and suggestions, which made the paper better. They are also grateful to the University of Nizwa, Sultanate of Oman, for providing all the necessary facilities and encouragement. REFERENCES [1] L. Herrera, J. Ospino and A. Di Parisco: All static spherically symmetric anisotropic solutions of Einstein’s equations, Phys. Rev. D 77 (2008), 027502. [2] R.C. Tolman: Static solutions of Einstein’s field equations for spheres of fluid, Phys. Rev. 55 (1939), 364. [3] H. A. Buchdahl: General relativistic fluid spheres, Phys. Rev. 116 (1959), 1027. [4] B. V. Ivanov: Gravity, Astrophysics, and String ‘02’, St. Kliment Ohridski University Press, Sofia 2003. [5] R. Ruderman: Pulsars: structure and dynamics, Annu. Rev. Astron. Astrophys. 10 (1972), 427. [6] V. Canuto: Neutron stars: general review, Solvay Conf. Astrophysics Gravitation, Brussels, Sept., 1973. [7] R. Kippenhahm and A. Weigert: Stellar Structure and Evolution, Springer, Berlin 1990. [8] A. I. Sokolov: Phase transitions in a superfluid neutron fluid, J. Exp. Theor. Phys. 79 (1980), 1137. [9] J. Binney and J. S. Tremaine: Galactic Dynamics, Princeton University Press, Princeton, NJ 1987. [10] P. Letelier: Anisotropic fluids with two-perfect-fluid components, Phys. Rev. D 22 (1980), 807. [11] S. S. Bayin: Anisotropic fluid spheres in general relativity, Phys. Rev. D 26 (1982), 1262. [12] R. L. Bowers and E. P. T. Liang: Anisotropic spheres in general relativity, Astrophys. J. 188 (1974), 657. [13] H. Heintzmann and W. Hillebrandt: Neutron stars with an anisotropic equation of state: mass, redshift and stability, Astron. Astrophys. 38, 51 (1975). [14] S. D. Maharaj and R. Maartens: Anisotropic spheres with uniform energy density in general relativity, Gen. Relat. Gravit. 21 (1989), 899. [15] M. K. Gokhroo and A. L. Mehra: Anisotropic spheres with variable energy density in general relativity, Gen. Relat. Gravit. 26 75, (1994). [16] M. S. R. Delgaty and K. Lake: Comput. Phys. Commun. 115, (1998), 395. [17] K. Dev and M. Gleiser: Gen. Relat. Gravit. 34, (2002), 1793.
40 [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48]
S. K. MAURYA, Y. K. GUPTA and M. K. JASIM K. Komathiraj and S. D. Maharaj: J. Math. Phys. 48, (2007), 042501. S. Thirukkanesh and F. C. Regel: Pramana - J. Phys. 78, (2012), 687. M.J. Sunzu, S. D. Maharaj and S. Ray: Astrophys. Space Sc. 352 (2014), 719–727. T. Harko and M. K. Mak: Anisotropic relativistic stellar models, Ann. Phys. 11 (2002), 3. M. K. Mak and T. Harko: Anisotropic stars in general relativity, Pr. Roy. Soc. A 459 393, (2003). M. Chaisi and S. D. Maharaj: Anisotropic static solutions in modelling highly compact bodies Pramana, J. Phys. 66 (2006), 609. S. K. Maurya and Y. K. Gupta: A family of anisotropic super-dense star models using a space-time describing charged perfect fluid distributions, Phys. Scr. 86 (2012), 025009. S. K. Maurya and Y. K. Gupta: Charged fluid to anisotropic fluid distribution in general relativity, Astrophys. Space Sci. 344 (2013), 243. S. K. Maurya and Y. K.Gupta: A new class of relativistic charged anisotropic super dense star Models, Astrophys. Space Sci. 353 (2014), 657. T. Feroze and A. A. Siddiqui: Gen. Relat. Gravit. 43 (2011), 1025. N. Pant, N. Pradhan and Newton K. Singh: J. Gravity (2014), 380320, 9 pages. N. Pant, N. Pardhan and M. Malaver: Anisotropic fluid star model in isotropic coordinates, Int. J. Astrophys. Space Sci. 3 (2015), 1–5. N. Pant, N. Pradhan and M. H. Murad: A class of super dense stars models using charged analogues of Hajj-Boutros type relativistic fluid solutions Int. J. Theor. Phys. 53 (2014), 3958. P. Bhara, F. Rahaman, S. Ray and V. Chatterjeed: Eur. Phys. J. C, Possibility of higher dimensional anisotropic compact star: arXiv: submit/1101295 [gr-qc] (2014). S. K. Hossein, F. Monowar, J. Rahaman, M. Naskar, M. Kalam and S. Ray, Int. J. Mod. Phys. D, arXiv:1204.3558v2 [gr-qc] (2012). M. Kalam, F. Rahaman, S. K. Hossein Monowar and S. Ray: Eur. Phys. J. C, Central Density Dependent Anisotropic Compact Stars, arXiv:1301.0271v1 [physics.gen-ph] (2012). M. Consenza, L. Herrera, M. Esculpi and E. Witten: Some models of anisotropic spheres in general relativity, J. Math. Phys. 22 (1981), 118. K.D. Krori, P. Bargohain and R. Devi: Some exact anisotropic solutions in general relativity, Can. J. Phys. 62 (1984), 239. T. Singh, G.P. Singh and R. S. Srivastava: Static anisotropic fluid spheres in general relativity with non uniform density, Int. J. Theor. Phys. 31 (1992), 545. L. K. Patel and N.P. Mehta: An exact model of an anisotropic relativistic sphere, Aust. J. Phys. 48 (1995), 635. M. Malaver: Regular model for a quark star with Van der Waals modified equation of state, World Applied Programming 3 (2013), 309. M. Malaver: Strange Quark model with Quadratic equation of state, Front. Math. Appl. 1 (2014), 9. M. Esculpi, M. Malaver and E. Aloma: A comparative analysis of the adiabatic stability of anisotropic spherically symmetric solutions, Gen. Relat. Gravit. 39 (2007), 633. L. Herrera and N. O. Santos: Local anisotropy in self-gravitating system, Phys. Rep. 53 (1997), 286. L. Herrera, N. O. Santos, A. Wang: Shearing expansion free spherical anisotropic fluid evaluation, Phys. Rev. D 78 (2008), 084026. L. Herrera et al.: Conformally flat anisotropic spheres in general relativity, J. Math. Phys. 42 (2001), 2129. L. Herrera et al.: Spherically symmetric dissipative anisotropic fluids: a general study, Phy. Rev. D 69 (2004), 084026. L. Herrera and J. Ponce de Leon: Isotropic and anisotropic charged spheres admitting a one-parameter group of conformal motions, J. Math. Phys. 26 (1985), 2302–2307. C. W. Misner and D. H. Sharp: Relativistic equations for adiabatic, spherically symmetric gravitational collapse, Phys. Rev. B 136 (1964), 571. L. Herrera: Cracking of self-gravitating compact objects, Phys. Lett. A 165 (1992), 206. H. Abreu, H. Hern´andez and L. A. N´un˜ ez: Sound speeds, cracking and the stability of self-gravitating anisotropic compact objects, Class. Quantum Gravit. 24 (2007), 4631.
REPORTS ON MATHEMATICAL PHYSICS
No. 1
RELATIVISTIC MODELLING OF STABLE ANISOTROPIC SUPER-DENSE STAR S. K. M AURYA1 , Y. K. G UPTA2 and M. K. JASIM1 1 Department
of Mathematical & Physical Sciences, College of Arts & Sciences, University of Nizwa, Nizwa- Sultanate of Oman 2 Department of Mathematics, Jaypee Institute of Information Technology University, Sector-128 Noida (U.P.), India (e-mails: [email protected], [email protected]; [email protected]; [email protected]) (Received November 24, 2014 – Revised May 12, 2015) In the present article we have obtained new set of exact solutions of Einstein field equations for anisotropic fluid spheres by using the Herrera et al. [1] algorithm. The anisotropic fluid solutions so obtained join continuously to the Schwarzschild exterior solution across the pressurefree boundary. It is observed that most of the new anisotropic solutions are well-behaved and are used to construct the super-dense star models such as neutron stars and pulsars. Keywords: anisotropic fluids, anisotropic factor, Einstein’s equations, Schwarzschild solution, neutron stars, pulsars.
1.
Introduction
The first ever exact solution of Einstein’s field equations for a compact object in static equilibrium was obtained by Schwarzschild in 1916. The static isotropic and anisotropic exact solutions describing stellar-type configurations have continuously attracted the interest of physicists like Herrera et al. [1, 41]. Tolman [2] has proposed an easy way to solve Einstein’s field equations by introducing an additional equation necessary to give a determinate problem in the form of some ad hoc relation between the components of metric tensor. According to this methodology, he obtained eight solutions of the field equations, and his important approach still is used in obtaining the exact interior solutions of the gravitational field equations for fluid spheres. Buchdahl [3] proposed a famous bound on the mass radius ratio of relativistic fluid spheres as 2GM/c2 r ≤ 8/9, which is an important contribution in order to study stability of the fluid spheres. Also Ivanov [4] has given the upper bound of the red shift for realistic anisotropic star models which cannot exceed the values of 3.842, provided the tangential pressure satisfies a strong energy condition (ρ ≥ pr + 2pt ) and when the tangential pressure satisfies the dominated energy condition (ρ ≥ pt ). [21]
22
S. K. MAURYA, Y. K. GUPTA and M. K. JASIM
Buchdahl [3] also obtained a nonsingular exact solution by choosing a particular choice of the mean density inside the star. The theoretical investigations of realistic fluid models indicate that stellar matter may be anisotropic at least in certain density ranges (ρ > 1015 gm/cm3 ) (Ruderman [5] and Canuto [6]) and radial pressure may not be equal to the tangential pressure of stellar structure. The existence of a solid core due to the presence of anisotropy in the pressure was thought of for type-3A super-fluid (Kippenhahm and Weigert [7]), different form of phase transitions (Sokolov [8]) or for other physical phenomena. On the scale of galaxies, Binney and Tremaine [9] have considered anisotropies in spherical galaxies, from a purely Newtonian point of view. The mixture of two gases (e.g. ionized hydrogen and electrons or monatomic hydrogen) can be described formally as an anisotropic fluid (Letelier [10] and Bayin [11]). The importance of equations of state for relativistic anisotropic fluid spheres have been investigated by generalizing the equation of hydrostatic equilibrium to include the effects of anisotropy (Bowers and Liang [12]). Their study shows that anisotropy may have nonnegligible effects on parameters such as maximum equilibrium mass and surface red shift. The relativistic anisotropic neutron star models with high densities by means of several simple assumptions showed that there is no limiting mass of neutron stars for arbitrary large anisotropy which have been studied by Heintzmann and Hillebrandt [13]. However, maximum mass of a neutron star still lies beyond 3–4 M⊙ . Also, the solutions for an anisotropic fluid sphere with uniform density and variable density have been studied by Maharaj and Maartens [14] and Gokhroo and Mehra [15], respectively. Most of the astronomical objects have variable density. Therefore, interior solutions of anisotropic fluid spheres with variable density are more realistic physically. Many workers have obtained different exact solutions for isotropic and anisotropic fluid spheres in different contexts (Delgaty and Lake [16], Dev and Gleiser [17], Komathiraj and Maharaj [18], Thirukkanesh and Ragel [19], Sunzu et al. [20], Harko and Mak [21], Mak and Harko [22], Chaisi and Maharaj [23], Maurya and Gupta [24–26], Feroze and Siddiqui [27], Pant et al. [28–30], Bhara et al. [31], Monowar et al. [32], Kalam et al. [33], Consenza et al. [34], Krori [35], Singh et al. [36], Patel and Mehta [37], Malaver [38, 39], Escupli et al. [40], Herrera and Santos [41, 42], Herrera et al. [43, 44]). The present paper consists of nine sections, Section 1 contains introduction; Section 2 contains metric, its components and the field equations. Section 3 embodies the solutions of anisotropic fluid spheres in different contexts. Section 4 contains the expressions for density and pressure which are mentioned for each fluid sphere. Section 5 describes various physical conditions to be satisfied by the anisotropic fluid spheres. The analytical behaviours of solutions under physical conditions (mentioned in Section 5) are mentioned in Section 6. Section 7 describes the evaluation of arbitrary constants involved in the fluid solutions by means of the smooth joining of Schwarzschild metric at the pressure-free interface r = a. The stability of models is proposed in Section 8, and finally Section 9 includes the physical analysis of the solutions so obtained along with the concluding remarks.
RELATIVISTIC MODELLING OF STABLE ANISOTROPIC SUPER-DENSE STAR
2.
23
Metric, components and field equations
The line element of a static spherical symmetric space-time in the curvature coordinates x i = (t, r, χ, ξ ) can be postulated as ds 2 = B 2 (r)dt 2 − ψ −1 (r)dr 2 − r 2 (dχ 2 + sin2 χ dξ 2 ),
(2.1)
while Einstein’s field equations are given as 1 j j j −κTi = Ri − Rδi , 2
(2.2)
where κ = 8πG/c4 . The components of the energy momentum tensor for spherically symmetric anisotropic fluid distribution are postulated in the form j
j
Ti = (c2 ρ + pt )vi v j − pt δi + (pr − pt )χi χ j ,
(2.3)
where v i is four-velocity B −1 v i = δ4i , χ i is the unit space-like vector in the direction √ of radial vector, χ i = ψδ1i , ρ is the energy density, pr is the pressure in direction of χ i (normal pressure) and pt is the pressure orthogonal to χi (transversal or tangential pressure). Suppose radial pressure is not equal to the tangential pressure, i.e. pr 6 = pt . If the radial pressure is equal to the transversal pressure, i.e. pr = pt , then it corresponds to isotropic or perfect fluid distribution. Let the measure of anisotropy be 1 = κ(pt − pr ), it is called the anisotropy factor (Herrera and Ponce de Leon [45]). The term 2(pt − pr )/r appears in the conservation equations Tji;i = 0 (where semi colon denotes the covariant derivative) which is representing a force due to anisotropic nature of the fluid. When pt > pr , then the direction of force is in the outward direction and inward when pt < pr . However, if pt > pr , then the force allows to construct a more compact object for anisotropic fluid than for isotropic fluid (Gokhroo and Mehra [15]). In view of the metric (2.1), the Einstein field equations (2.2) give 1−ψ ψ′ 8πG 2B ′ ψ ψ −1 8πG ρ = − , p = + , (2.4) r 2 2 4 c r r c Br r2 ′ ′′ 1 B B′ 1 1 ′ B ψ + + 2ψ − − 2 =2 1− 2 , (2.5) B r B rB r r where the “dash” denotes the derivative with respect to r. 3.
Classes of solutions
The field Eq. (2.5) has two dependent variables B(r) and ψ(r), therefore Eq. (2.5) can admit infinitely many solutions for different choices of B(r) and ψ(r) but all these solutions may or may not satisfy the physical conditions for the fluid spheres. For a given B(r), Eq. (2.5) reduces to a first-order ordinary differential equation in ψ(r). For its physically valid solution we will have to choose the metric potential
24
S. K. MAURYA, Y. K. GUPTA and M. K. JASIM
B(r) such that B(0) is positive and finite. This is a sufficient condition for a static fluid sphere to be regular at the centre. Let us take B(r) of the form B(r) = D(1 − c0 r 2 )n , where n 6 = 0 and D is a positive arbitrary constant. (3.1)
Maurya and Gupta [22, 23] have already obtained all possible anisotropic solutions of Einstein field equations for n a positive integer ≥ 1 with c0 < 0, and for −1 < n < 0, c0 > 0, with different anisotropic factor 1. But in the present problem we consider B(r) = D(1 − c0 r 2 )n with n = −1, −2 and −3 and the anisotropy factor
10 c0 (c0 r 2 )−n ; (3.2) [1 − (n + 1)c0 r 2 ]1+n where c0 and 10 are positive constants. Herrera et al. [1] have proposed an algorithm for all possible spherically symmetric anisotropic solutions of Einstein field equations. By using the Herrera et al. [1] algorithm, Eq. (2.5) reduces to ′ 2y 6 4 2 1 ψ′ + + 2y − + 2 ψ= 1− 2 , (3.3a) y r r y y r where B ′ (r) 1 y(r) = + . (3.3b) 2B(r) r Integrating (3.3a) we can obtain ψ as " Z # R 2 y(r)(1 + 1(r)r 2 )e [(4/r y(r))+2y(r)]dr 6 r −2 dr + A r8 R ψ= , (3.3c) 2 y 2 (r)e [(4/r y(r))+2y(r)]dr and [1 − (n + 1)c0 r 2 ] y(r) = . (3.3d) r(1 − c0 r 2 ) Further Z −2φ Z −2φ e e 3 2φ ψ = φf e 10 dφ − dφ + Aφf 3 e2φ for n = −1, (3.4a) 2 f φf 2 and Z Z φf 2−n f n−1 f n−1 φf 2−n ψ = 2/(n+1) 10 dφ − dφ + A ; g φ n+1 g n(n+3)/(n+1) φ 2 g (n−1)/(n+1) g 2/(n+1) for n 6= −1, (3.4b) 1=
where φ = c0 r 2 , f = (1 − φ) and g = [1 − (n + 1)φ].
RELATIVISTIC MODELLING OF STABLE ANISOTROPIC SUPER-DENSE STAR
25
Eqs. (3.4a) and (3.4b) give the following solutions: ψ = Aφf 3 e2φ + (1 − 2φ)f 2 − (6 + 10 )f 3 φe2φ−2 Ei(2 − 2φ) + 10 φf 2 ; for n = −1, (3.5a) where Ei(2 − 2φ) = log(2 − 2φ) +
∞ P
N =1
(2−2φ)N N!N
,
4 2 f 15 2 4 2 (15φ − 25φ + 8) 2 4 ψ = Aφ(1 + φ) f + f + φ(1 + φ) f log 8 16 1+φ 10 1−φ + φf 2 2(φ 2 − φ + 2) + f 2 (1 + φ)2 log ; for n = −2, 16 1+φ (3.5b) 1 5 3h 1 + 2φ ψ = Aφf 5 (1 + 2φ) − f − 320φ log (1 + 2φ) 3 243 f f + 10 φf 3 (1 + 2φ)(1 − 3φ + 3φ 2 ) ; for n = −3, (3.5c) where h = (−81 + 130φ + 36φ 2 − 720φ 3 + 320φ 4 ). 4.
Expressions for energy density, pressures for different values of n
(a) For n = −1,
8πG ρ c2
8πG pr c4
8πG pt c4
= c0 [Af 2 e2φ (4φ 2 + 5φ − 1) + (6 + 10 )f 2 (3 − 5φ − 4φ 2 )e2φ−2 Ei(2 − 2φ) + (6 − 11φ 2 + 2φ 3 ) + 10 [(−3 + 10φ − 7φ 2 ) ¯ − 2φ)]], + 2φe2φ−2 f 3 Ei(2
(4.1a)
= c0 [Af 2 (1 + φ)e2φ + (2 − 10φ + 7φ 2 − 2φ 3 ) − 6f 2 (1 + 5φ)e2φ−2 Ei(2 − 2φ)
+10 (1 − φ 2 )[1 − e2φ−2 f Ei(2 − 2φ)]],
(4.2a)
= c0 [Af 2 (1 + φ)e2φ + (2 − 10φ + 7φ 2 − 2φ 3 ) − 6f 2 (1 + 5φ)e2φ−2 Ei(2 − 2φ)
+10 [(1 + φ − φ 2 ) − (1 − φ 2 )e2φ−2 f Ei(2 − 2φ)]]. ¯ where Ei(2 − 2φ) =
−2 (2−2φ)
−2
∞ P
N=1
(2−2φ)N−1 . N!
(4.3a)
26
S. K. MAURYA, Y. K. GUPTA and M. K. JASIM
(b) For n = −2,
8πG ρ c2
48+85φ−350φ 2 +90φ 3 +330φ 4 −195φ 5 +30φf 3 (1+φ) 8 1 3 1−φ 2 3 3 2 − f 45−15φ−285φ −225φ −10 (15φ +19φ +φ−3) log 16 1+φ 10 f [11φ 3 −16φ 2 +19φ−6+2f 2 φ(1+φ)] − 8
= c0
+Af 3 (−3+φ+19φ 2 +15φ 3 ) , (4.1b) 8πG 1 2 3 2 3 4 5 p = c A(1+3φ)f (1+φ) + (16−49φ−50φ +90φ +30φ −45φ ) r 0 c4 8 1−φ (15+10 ) 3 2 f (1+3φ)(1+φ) log + 16 1+φ 10 (4.2b) + f (1+3φ)(4−2φ+2φ 2 ) , 16 8πG (16−49φ−50φ 2 +90φ 3 +30φ 4 −45φ 5 ) 3 2 = c A(1+3φ)f (1+φ) + p 0 t c4 8 (15+10 ) 3 1−φ + f (1+3φ)(1+φ)2 log 16 1+φ 10 (4.3b) + (4+6φ+2φ 2 +26φ 3 −6φ 4 ) . 16 (c) For n = −3,
8πG ρ c2
= c0 −A(3−15φ+90φ 3 −165φ 4 +117φ 5 −34φ 6 ) 1 (21320−26565φ−55034φ 2 +156004φ 3 −11040φ 4 +29120φ 5 ) 81 320 4 1+2φ + f (−1+5φ+26φ 2 ) log 243 1−φ 2 2 3 4 −10 f (3−14φ−10φ +93φ −90φ ) , (4.1c) +
8πG pr c4
= c0 Af 4 (1+7φ+10φ 2 )−(−356+411φ+146φ 2−3820φ 3−1680φ 4 +1600φ 5 ) 320 4 1+2φ + f (1+2φ)(1+5φ) log 243 1−φ 2 2 3 4 +10 f (1+4φ−8φ −9φ +30φ ) , (4.2c)
27 = c0 Af 4 (1+7φ+10φ 2 )+(356−411φ−146φ 2 +3820φ 3 +1680φ 4−1600φ 5 ) 320 4 1+2φ + f (1+2φ)(1+5φ) log 243 1−φ 2 2 3 4 3 2 +10 f (1+4φ−8φ −9φ +30φ )+φ (1+2φ) . (4.3c)
RELATIVISTIC MODELLING OF STABLE ANISOTROPIC SUPER-DENSE STAR
5.
8π G pt c4
Reality and physical (well-behaved) conditions for anisotropic solutions
The physically meaningful anisotropic solution for the Einstein’s field equations must satisfy some physical conditions (Mak and Harko [20], Maurya and Gupta [22]) such as: (i) The solution should be free from physical and geometrical singularities, i.e. the pressure and energy density should be finite at the centre and the metric potentials B(r) and ψ(r) must have nonzero positive values. (ii) The radial pressure pr must be vanishing but the tangential pressure pt may not vanish at the boundary r = ra of the fluid sphere. However, the radial pressure is equal to the tangential pressure at the centre of the fluid sphere. (iii) The density ρ, radial pressure pr and tangential pressure pt should be positive inside the star. (iv) (dpr /dr)r=0 = 0 and (d 2 pr /dr 2 )r=0 < 0, so that the pressure gradient dpr /dr is negative for 0 ≤ r ≤ ra . (v) (dpt /dr)r=0 = 0 and (d 2 pt /dr 2 )r=0 < 0, so that the pressure gradient dpt /dr is negative for 0 ≤ r ≤ ra . (vi) (dρ/dr)r=0 = 0 and (d 2 ρ/dr 2 )r=0 < 0, so that the density gradient dρ/dr is negative for 0 ≤ r ≤ ra . The conditions (iv)–(vi) imply that the pressure and density should be maximal at the centre and monotonically decreasing towards the surface. (vii) Inside the fluid ball the speed of sound should be less than that of light, i.e. s s dpr dpt < 1, 0≤ < 1. 0≤ 2 c dρ c2 dρ In addition, the velocity of sound should be monotonically decreasing away from the centre, i.e. 2 2 d pr d dpt d pt d dpr < 0 or > 0 and < 0 or >0 2 dr dρ dρ dr dρ dρ 2
for 0 ≤ r ≤ ra . In this context, it is worth mentioning that the equation of state at ultra-high distribution has the property that the speed of sound is decreasing outwards (Canuto [30]). (viii) A physically reasonable energy-momentum tensor has to obey the energy conditions ρ ≥ pr + 2pt (strong energy condition) and ρ + pr + 2pt ≥ 0.
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S. K. MAURYA, Y. K. GUPTA and M. K. JASIM
(ix) The red shift at center Z0 and at surface Za should be positive, finite and both bounded. (x) The anisotropy factor 1 should be zero at the center and must be increasing towards the surface. 6.
Physical properties of the new solutions (a) For n = −1 The expressions for pressure and density at the centre are given as: 10 e2 − Ei(2)(6 + 10 ) 8πG 8πG = = c0 A + 2 + , pr pt c4 c4 e2 r=0 r=0 8πG Ei(2)(18 + 310 ) − 3e2 10 ρ = c0 −A + 6 + . c2 e2 r=0
(6.1a) (6.2a)
The pressure and density should be positive at the centre and consequently A satisfies the following inequalities Ei(2)(6 + 10 ) − 10 e2 Ei(2)(18 + 310 ) − 310 e2 and A < 6 + . A > −2 + e2 e2 (6.3a) The ratio of pressure-density should be positive and less than 1 at the centre, i.e. p0 /ρ0 c2 ≤ 1 which gives the following inequality, pr pt e2 (A + 2) + 10 e2 − Ei(2)(6 + 10 ) = = ≤ 1. (6.4a) ρc2 r=0 ρc2 r=0 e2 (−A + 6) + Ei(2)(18 + 310 ) − 310 e2
From Eq. (6.4a) we get
Ei(2)(6 + 10 ) − 210 e2 . e2 Using Eqs. (6.3a) and (6.5a), A satisfies the following inequalities Ei(2)(6 + 10 ) − 210 e2 Ei(2)(6 + 10 ) − 10 e2
(6.5a)
10 ≥ 0. (6.6a)
Differentiating (4.2a) with respect to r, we get an expression for the gradient of pressure h 8πG dpr 2 r Ae2φ (1 − 4φ + φ 2 + 2φ 3 ) + (−10 + 14φ − 6φ 2 ) = 2c 0 c4 dr ¯ − 6f 2 (1 + 5φ)e2φ−2 Ei(2 − 2φ) − 6(5 − 12φ − 3φ 2 + 10φ 3 )e2φ−2 Ei(2 − 2φ)
+ 10 [−2φ + e2φ−2 Ei(2 − 2φ)(−2 + 5φ − 3φ 3 ) i ¯ − e2φ−2 f 2 (1 + φ)Ei(2 − 2φ)] .
(6.7a)
RELATIVISTIC MODELLING OF STABLE ANISOTROPIC SUPER-DENSE STAR
29
Thus it is found that the extremum of pr occurs at the centre, i.e. and
pr′ = 0 ⇒ r = 0 2c02 8πG ′′ ¯ ¯ p = − (10 − A)e2 + 6Ei(2) + 30Ei(2) + 10 [2Ei(2) + Ei(2)] . r r=0 4 2 c e (6.8a)
This shows that the expression on the right-hand side of Eq. (6.8a) is negative for all values of A and 10 satisfying condition (6.8a). Then the radial pressure (pr ) has maximum at the centre and is monotonically decreasing. Differentiating (4.3a) with respect to r we get h 8π G dpt 2 2φ 2 3 2 = 2c r 0 Ae (1 − 4φ + φ + 2φ ) + (−10 + 14φ − 6φ ) + 10 (1 − 2φ) c4 dr + e2φ−2 Ei(2 − 2φ)[10 (−2 + 5φ − 3φ 3 ) − 6(5 − 12φ − 3φ 2 + 10φ 3 )] i ¯ − (6 + 10 )f 2 (1 + 5φ)e2φ−2 Ei(2 − 2φ) , (6.9a)
this suggests that the extremum value of pt occurs at the centre, i.e.
pt′ = 0 ⇒ r = 0 (6.10a) and 2c02 8π G ′′ 2 ¯ ¯ p = − (10 − A − 1 )e + 6 Ei(2) + 30Ei(2) + 1 [2Ei(2) + Ei(2)] . 0 0 t r=0 c4 e2 (6.11a) Under the condition (6.6a), the expression on the right-hand side of Eq. (6.11a) is negative for all values of A and 10 . This shows that the tangential pressure (pt ) is maximum at the centre and it is monotonically decreasing. Now, differentiating Eq. (4.1a) with respect to r, we get h 8πG dρ 2 2φ 2 3 4 = 2c r 0 Ae (5 − 23φ + 10φ + 8φ ) c2 dr + (−30 − 24φ − 60φ 3 − 48φ 4 )e2φ−2 Ei(2 − 2φ) ¯ + 6f 2 (3 − 5φ − 4φ 2 )e2φ−2 Ei(2 − 2φ) − 22φ + 6φ 2
where
+ 10 [e2φ−2 f {Ei(2 − 2φ)(−5 − 19φ + 12φ 2 + 8φ 3 ) ¯ − f (−5 + 9φ + 8φ 2 )Ei(2 − 2φ) i ¯¯ − 2f 2 φ Ei(2 − 2φ)} − 2(−5 + 7φ)] , (6.12a)
¯¯ Ei(2 − 2φ) =
∞ X N (2 − 2φ)N −1 4 + 4 . (2 − 2φ)2 (N + 1)! N=1
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S. K. MAURYA, Y. K. GUPTA and M. K. JASIM
The extremum of ρ occurs at the centre, i.e. ρ′ = 0 ⇒ r = 0
and
2c02 8πG ′′ ¯ ρ = − −(5A + 1010 )e2 + Ei(2)(30 + 510 ) − Ei(2)(18 + 510 ) . r=0 2 2 c e (6.13a)
Thus, the expression on the right-hand side of (6.12a) is negative. This shows that the density ρ has maximum at the centre and is monotonically decreasing towards the pressure-free interface. 1 dp The square of adiabatic sound speed at the centre, 2 , is given by c dρ r=0 ¯ ¯ 1 dpr (10 − A)e2 + 6Ei(2) + 30Ei(2) + 10 [2Ei(2) + Ei(2)] = , (6.14a) ¯ c2 dρ r=0 (5A + 1010 )e2 + Ei(2)(30 + 510 ) + Ei(2)(18 + 510 ) ¯ ¯ 1 dpt (10 − A − 10 )e2 + 6Ei(2) + 30Ei(2) + 10 [2Ei(2) + Ei(2)] . = ¯ c2 dρ r=0 (5A + 1010 )e2 + Ei(2)(30 + 510 ) + Ei(2)(−18 + 510 ) (6.15a) The causality condition is obeyed at the centre for all values of constants under the condition (6.6a), i.e. the radial and tangential velocities of sound are monotonically decreasing and less than 1. (b) For n = −2
The central values of pressure and density are given by 8πGpt c0 8πGpr = = (4A + 8 + 10 ) , 4 4 c c 4 r=0 r=0 8πGρ c0 = (−12A + 24 + 310 ) . 2 c 4 r=0
(6.1b) (6.2b)
The central values of pressure and density should be positive and finite, then A satisfies the following conditions: 10 10 A > −2 − and A < 2 − . (6.3b) 4 4 The condition that the ratio of pressure and density should be positive and less than 1 at the centre, i.e. ρpc02 ≤ 1 leads to the inequality 0 pr pt 4A + 8 + 10 = = ≤ 1. (6.4b) ρc2 r=0 ρc2 r=0 −12A + 24 + 310 From inequality (6.4b) we get
10 A≤ 1+ . 8
(6.5b)
RELATIVISTIC MODELLING OF STABLE ANISOTROPIC SUPER-DENSE STAR
31
Owing to inequalities (6.3b) and (6.5b), A satisfies the inequality 10 10
64 + 130x − 330x 2 − 150x 3 + 270x 4 8 10 + (2 − 16φ + 19φ 2 −10φ 3 −3φ 4 ) . (6.7b) 8 Thus, it is found that the extremal value of pr occurs at the centre, i.e. 8πG ′′ −16A − 64 + 210 ′ pr = 0 ⇒ r = 0 and pr r=0 = . (6.8b) c4 8 Hence, the expression on the right-hand side of Eq. (6.8b) is negative for all values of constants A and 10 satisfying condition (6.6b) and it shows that the pressure pr has maximum at the centre and is monotonically decreasing. Differentiating Eq. (4.3b) with respect to r we get 1 − φ 8πG dpt 15 − 10 2 f 2 (−1 + 3φ + 13φ 2 + 9φ 3 ) = 2c0 r −2A + log c4 dr 8 1 + φ −
64 + 130x − 330x 2 − 150x 3 + 270x 4 8 10 2 3 4 (6.9b) + (2 + 43φ −10φ −3φ ) , 8 Thus, it is found that the extremal value of pt occurs at the centre, i.e. 8πG ′′ 2 16A − 64 + 210 ′ pt = 0 ⇒ r = 0 and pt r=0 = 2c0 . (6.10b) c4 8 The expression on the right-hand side of Eq. (6.10b) is also negative for all values of A and 10 satisfying condition (6.6b). This behaviour shows that the transversal pressure pt has maximum at the centre and is monotonically decreasing. Now, differentiating Eq. (4.1b) with respect to r, the expression for the gradient of density is 8π G dρ 5S 15+10 1−φ 2 = 2c r − 2A+ log f 2 (−5−17φ +25φ 2 +45φ 3 ) 0 c2 dr 8(1−φ 2 ) 8 1+φ 10 2 3 3 − (−30+88φ −72φ +2φ +25φ ) , (6.11b) 8 −
where S = (32 + 86φ + 10φ 2 − 176φ 3 − 312φ 4 + 90φ 5 + 273φ 6 ).
32
S. K. MAURYA, Y. K. GUPTA and M. K. JASIM
Thus, the extremal value of ρ occurs at the centre, i.e. 8πG ′′ ′ 2 180 + 80A + 3010 ρ r=0 = 2c0 ρ = 0 ⇒ r = 0 and > 0. c2 8
(6.12b)
The right-hand side of Eq. (6.12b) is positive due to the inequality (6.6b), and this condition gives that the energy density ρ is minimal at the centre and is monotonically increasing. 1 dp Also, the square of its adiabatic sound speed at the centre, 2 , is given c dρ r=0 by 1 dpt 1 dpr (16A − 64 + 310 ) = 2 = . (6.13b) 2 c dρ r=0 c dρ r=0 [180 + 80A + 3010 ]
The causality condition is negative at the centre for all values of constants satisfying condition (6.6b). Due to the increasing nature of energy density, the solution behaves not well for n = −2. Case 3: n = −3.
The central values of pressure and density are given by 8πGpt 8πGpr = = c0 [A + 356 + 10 ] , c4 c4 r=0 r=0 8πGρ c0 = [−243A + 21320 − 24310 ] . c2 81 r=0
(6.1c) (6.2c)
The central values of pressure and density should be positive and finite. Then A satisfies the following conditions: 21320 A > (−356 − 10 ) and A < − 10 . (6.3c) 243 The condition that the ratio of pressure-density is positive and less than 1 at the centre, i.e. ρpc02 ≤ 1, leads to the inequality 0 pr pt 81 [A + 356 + 10 ] = = ≤ 1. (6.4c) ρc2 r=0 ρc2 r=0 −243A + 21320 − 24310 Eq. (6.4c) leads to
7516 A≤− + 10 . 324
Using Eqs. (6.3c) and (6.5c) we get the inequalities for A as 7516 − (356 + 10 ) < A ≤ − + 10 . 324
(6.5c)
(6.6c)
33
RELATIVISTIC MODELLING OF STABLE ANISOTROPIC SUPER-DENSE STAR
Differentiating (4.2c) with respect to r we get 8πG dpr 2 = 2c0 r Ap1 (φ) − (411 − 292φ − 11460φ 2 − 6720φ 3 + 8000φ 4 ) c2 dr 320 3 320 1 + 2φ + f (1 + 5φ) + p2 (φ) log 243 243 1−φ + 10 (2 − 30φ + 33φ 2 + 160φ 3 −345x 4 + 180φ 5 ) ,
where
(6.7c)
p1 (φ) = (3 − 24φ−6φ 2 + 132φ 3 −165φ 4 + 60φ 5 ),
p2 (φ) = (15 − 40φ − 252φ 2 − 336φ 3 + 120φ 4 ).
Thus, it is found that the extremal value of pr occurs at the centre, i.e. 8πG ′′ 320 2 p = 2c + 21 (6.8c) pr′ = 0 ⇒ r = 0 and 3A − 411 + 0 . r r=0 0 c4 243 So, the expression on the right-hand side of Eq. (6.8c) is negative for all values of A and 10 satisfying condition (6.6c). This shows that the pressure (pr ) has maximum at the centre and is monotonically decreasing. Differentiating (4.3c) with respect to r we get 8π G dpt 2 = 2c r A(3 − 24φ−6φ 2 + 132φ 3 −165φ 4 + 60φ 5 ) 0 c2 dr 320 3 − (411 − 292φ − 11460φ 2 − 6720φ 3 + 8000φ 4 ) + f (1 + 5φ) 243 320 1 + 2φ + (15 − 40φ − 252φ 2 − 336φ 3 + 120φ 4 ) log 243 1−φ (6.9c) + 10 (2 − 30φ + 36φ 2 + 176φ 3 −325x 4 + 180φ 5 ) . Thus, it is found that the extremum of pt occurs at the centre, i.e. 320 8πG ′′ 2 ′ + 210 . pt = 0 ⇒ r = 0 and 4 pt r=0 = 2c0 3A − 411 + c 243
(6.10c)
It is clear that the expression on the right-hand side of Eq. (6.10c) is negative for all values of A and 10 satisfying condition (6.6c). This condition gives that the transversal pressure is maximal at the centre and is monotonically decreasing. Now, differentiating equation (4.1c) with respect to r we get ρ1 8πG dρ 2 = 2c0 r A(15 − 270x 2 + 660x 3 − 585x 4 + 204x 5 ) + 2 c dφ 81 320 1 + 2φ 3 4 5 + (9 − 140x + 540x − 495x + 156x ) log 81 1−φ + 10 (2 − 30φ + 33x 2 + 160φ 3 −345x 4 + 180φ 5 ) , (6.11c)
34
S. K. MAURYA, Y. K. GUPTA and M. K. JASIM
ρ1 = (−26885−55018φ +64686φ 2 −1149772φ 3 +685612φ 4 −296960φ 5 +128960φ 6 ). Thus, the extremum of ρ occurs at the centre, i.e. 8πG ′′ 2 −26885 ρ ′ = 0 ⇒ r = 0 and ρ = 2c + 15A + 21 0 , 0 r=0 c2 81
(6.12c)
Hence, the expression on the right-hand side of Eq. (6.12c) is negative for all values of A and 10 satisfying condition (6.6c). Then the density ρ has maximum at the centre and is monotonically decreasing. , is given by The square of the adiabatic sound speed at the centre, c12 dp dρ r=0 1 dpt −99553 − 729A + 48610 1 dpr = = . (6.13c) c2 dρ r=0 c2 dρ r=0 3[−26885 + 1215A + 16210 ]
The causality condition is less than 1 and positive at the centre for all values of A and 10 satisfying condition (6.6c). 7.
Boundary conditions for evaluation of constants A, D and c0 The above system of equations is to be solved subject to the boundary condition that the radial pressure pr = 0 at r = a (where r = a is the outer boundary of the fluid sphere). It is clear that m(r = a) = M is a constant and, in fact, the interior metric (2.1) can be joined smoothly at the surface of sphere (r = a) to an exterior Schwarzschild metric whose mass is the same as above, i.e. m(r = a) = M (Masiner and Sharp [45]). Then the interior metric of this fluid sphere should be joined smoothly with Schwarzschild exterior metric such as B 2 (a) = 1 − 2u, where u = M/a, where M is the mass of the fluid sphere as measured by the exterior field and a is the boundary of the sphere. By joining (3.5a) and (3.5c) on the boundary of the anisotropic fluid sphere (r = a) and by setting φa = co a 2 and ψanis (a) = 1 − 2uanis , we get the expressions for mass for n = −1 and −3 as: a Manis = φa − Afa3 e2φa + (4φa − 3φa2 + 2φa3 ) 2 Manis =
+ (6 + 10 φa )fa3 e2φa −2 Ei(2 − 2φa ) − 10 φa fa2 ,
a M1 φa − A(1 − φa )5 (1 + 2φa ) 2 81 320 1 + 2φa 3 5 + (1 − φa ) (1 + 2φa ) log − M2 10 φa .fa , 243 1 − φa
M1 = (292 − 305φa − 662φa2 + 1796φa3 − 1360φa4 + 320φa5 ), M2 = (1 + 2φa )(1 − 3φa + 3φa2 ),
S = (16 + 17φa − 50φa2 + 10φa3 + 30φa4 − 15φa5 ).
(7.1)
(7.2)
35
RELATIVISTIC MODELLING OF STABLE ANISOTROPIC SUPER-DENSE STAR
The arbitrary constant A in (3.5a) and (3.5c) can be determined by putting the radial pressure pr equal to zero at the boundary, the expressions of constant A for n = −1 and −3 are given as: [6(1 + 5φa ) + 10 fa ]fa2 e2φa −2 Ei(2 − 2φa ) − (2 − 10φ + 7φ 2 − 2φ 3 ) − 10 fa2 ; (7.3) fa2 (1 + φa )e2φa (−356 + 411φa + 146φa2 − 3820φa3 − 1680φa4 + 1600φa5 ) 320 4 1 + 2φa 2 2 3 4 − − 10 fa (1 + 4φ − 8φ − 9φ + 30φ ) fa (1 + 2φa )(1 + 5φa ) log 243 1 − φa A= . fa4 (1 + 7φa + 10φa2 ) (7.4) A=
Also, the arbitrary constants D in the metric potential for the cases n = −1 and −3 can be computed by the condition B(a) = 1 − 2uanis , D = fa3 [Aφa fa e2φa + (1 − 2φa ) − (6 + 10 )fa φa e2φa −2 Ei(2 − 2φa ) + 10 φa ]; 1 + 2φa 1 2 3g (1 + 2φ ) fa D = fa6 Aφa fa2 (1 + 2φa ) − − 320φ log a a 243 fa3 fa +10 φa (1 + 2φa )(1 − 3φa + 3φa2 ) .
(7.5)
(7.6)
The positive constant c0 can be calculated 2 × 1014 by taking the surface density gm/cm3 and using the condition c0 = 8πc2G a 2 ρa / 1 − ψa − aψa′ for n = −1 and −3. Tables for numerical values of physical quantities
In Tables 1–2: Z = red shift, solar mass M⊙ = 1.475 km, G = 6.673 × 10 cm3 /gs2 , c = 2.997 × 1010 cm/s, D = (8π G/c2 c0 )ρ, Pr = (8π G/c4 c0 )pr a 2 , Pt = (8πG/c4 c0 )pt a 2 . −8
Table 1. n = −1, 10 = 4.2395, c0 a 2 = 0.0829, radius (a) = 16.0780 km, mass (M) = 1.7609 M⊙ . p p r/a Pr Pt D 1 dpr /c2 dρ dpt /c2 dρ pr /c2 ρ pt /c2 ρ Z 0.0
1.3165
1.3165
20.7045
0.0000
0.9999
0.9394
0.0636
0.0636
0.2570
0.2
1.2543
1.2683
20.5763
0.0140
0.9918
0.9312
0.0610
0.0616
0.2554
0.4
1.0723
1.1285
20.1933
0.0562
0.9680
0.9070
0.0531
0.0559
0.2504
0.6
0.7852
0.9117
19.5603
0.1265
0.9292
0.8672
0.0401
0.0466
0.2421
0.8
0.4167
0.6417
18.6860
0.2249
0.8761
0.8125
0.0223
0.0343
0.2303
1.0
0.0000
0.3515
17.5839
0.3515
0.8092
0.7427
0.0000
0.0200
0.2151
36
S. K. MAURYA, Y. K. GUPTA and M. K. JASIM
Table 2. n = −3, 10 = 0.7094, c0 a 2 = 0.0028, radius (a) = 3.1274 km, mass (M) = 0.8672 M⊙ . p p r/a Pr Pt D 1 × 1013 dpr /c2 dρ dpt /c2 dρ pr /c2 ρ pt /c2 ρ Z 0.0 4.0619
4.0619
13.1901
0
0.5160
0.5160
0.3079
0.3079
0.8597
0.2 3.8982
3.8982
13.1838
0.00997
0.5158
0.5158
0.2957
0.2957
0.8595
0.4 3.4077
3.4077
13.1650
0.63900
0.5153
0.5153
0.2588
0.2588
0.8590
0.6 2.5922
2.5922
13.1336
7.2949
0.5144
0.5144
0.1974
0.1974
0.8582
0.8 1.4549
1.4549
13.0896
41.116
0.5132
0.5132
0.1111
0.1111
0.8571
1.0 0.0000 1.5748*10−8 13.0331
157.48
0.5116
0.5116
0.0000
1.208*10−9 0.8556
8.
Stability of the stellar structure For physically acceptable model, one aspect is that the velocity of sound should be within the range 0 ≤ va2 = (dp/c2 dρ) ≤ 1 (Herrera [47] and Abreu et al. [48]). In the present models, the expression for velocity of sound at the centre is given by equations (6.13a), (6.14a) and (6.13c). We plot the radial and transversal velocities of sound in Fig. 3 and conclude that all parameters satisfy the inequalities 2 2 = (dpr /c2 dρ) ≤ 1 and 0 = vat = (dpt /c2 dρ) ≤ 1, everywhere inside the star. 0 = var 2 2 From Eqs. (6.13a), (6.14a) and (6.13c), we found that |vat − var | ≤ 1 at the centre and proved that the velocity of sound is monotonically decreasing inside the star. 2 2 2 2 Also 0 ≤ vat ≤ 1 and 0 ≤ var ≤ 1, therefore vat − var ≤ 1. Now, to examine the stability of local anisotropic fluid distribution we use Herrera’s [46] concept. Herrera [46] proposed the cracking (also known as overturning) concept which the states that the region, in which radial speed of the sound is greater than the transversal speed is a potentially stable region. In our proposed models, the models are stable with the radius 16.0780 km, 10 = 4.2395, c0 a 2 = 0.0829 for n = −1 and radius 3.1274 km, 10 = 0.7094, c0 a 2 = 0.0028 for n = −3. 9.
Physical analysis and conclusions In the present paper, we have presented the new set of anisotropic exact solutions of Einstein’s field equations by taking the metric potential g44 = D(1 − c0 r 2 )n for n = −1, −2 and −3 and the specific choice of anisotropic factor 1 which involves the anisotropic parameter 10 . The solutions so obtained are used to construct the super dense star models with the surface density 2 × 1014 gm/cm3 . It is observed that the solutions satisfy all reality and physical conditions (mentioned in Section 5) for n = −1 and −3. But the solution is not compatible for n = −2 due to the increasing nature of its density (Section 6, Case (b)). The anisotropic fluid sphere possesses the maximum mass and corresponding radii are 1.7609 M⊙ and 16.0780 km for n = −1, c0 a 2 = 0.0829, 10 = 4.2395 and 0.8672 M⊙ and 3.1274 km for n = −3, c0 a 2 = 0.0028, 10 = 0.7094. The red shift for n = −1 and −3 are monotonically decreasing towards the pressure free interface r = a and we have found that the
RELATIVISTIC MODELLING OF STABLE ANISOTROPIC SUPER-DENSE STAR
37
18 16 14 12 10 8
n = −1 PrPrforforn=-1 D − Pr − D-Pr-2Pt for2P n=-1 t for n = −1
PrPrforforn=-3 n = −3 D − Pr − D-Pr-2Pt for2P n=-3 t for n = −3
6 4 2 0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
r/a Fig. 1. Variations of radial pressure Pr and the trace D − Pr − 2Pt of the energy-momentum tensor for n = −1, c0 a 2 = 0.0829, 10 = 4.2395 and n = −3, c0 a 2 = 0.0028, 10 = 0.7094.
1 0.9
Vr Vr for forn=-1 n = −1
0.8
R forn=-1 n = −1 Rrr for
0.7 0.6 0.5
Vr for forn=-3 n = −3 Vr
Rrr for R forn=-3 n = −3
0.4 0.3 0.2
0.1 0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
1
r/a p Fig. 2. Variations of the radial velocity Vr = dpr /c2 dρ and ratio of radial pressure and density Rr = pr /c2 ρ of the energy-momentum tensor for n = −1, c0 a 2 = 0.0829, 10 = 4.2395 and n = −3, c0 a 2 = 0.0028, 10 = 0.7094.
red shift at the centre (Z0 ) and at the surface (Za ) are: (i) Z0 = 0.2570 and Za = 0.2151 for n = −1, (ii) Z0 = 0.8597 and Za = 0.8556 for n = −3 for both strong energy and dominated energy conditions. In our models, the red shift also satisfied the upper bound limit for the realistic anisotropic star models (Ivanov [4])
38
S. K. MAURYA, Y. K. GUPTA and M. K. JASIM
1 0.9 0.8
Vt Vt for forn=-1 n = −1
0.7
Rt for forn=-1 n = −1 Rt
0.6
Vt Vt for forn=-3 n = −3
0.5
Rt Rt for forn=-3 n = −3
0.4 0.3 0.2 0.1 0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
1
r/a p Fig. 3. Variations of the tangential velocity Vt = dpt /c2 dρ and the ratio of tangential pressure and density Rt = pt /c2 ρ of the energy-momentum tensor for n = −1, c0 a 2 = 0.0829, 10 = 4.2395 and n = −3, c0 a 2 = 0.0028, 10 = 0.7094. 42
anisotropic anisotropy factor (delta1) (11 ) forfornn=-1 = −1
36
anisotropyfactor factor(delta2) (12 ) for = −3 ansotropy for n n=-3
30 24 18
12 6 0 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
1
r/a Fig. 4. Variations of the anisotropy factor 11 = 1 × 10−2 and anisotropy factor 12 = 1 × 10−9 of the energy-momentum tensor for n = −1, c0 a 2 = 0.0829, 10 = 4.2395 and n = −3, c0 a 2 = 0.0028, 10 = 0.7094.
and its behaviour is represented by Fig. 5. The Tables 1 and 2 show the values of physical parameters. Fig. 1 shows that the fluid spheres satisfies energy condition. The behaviours of velocity and pressure density ratios by Figs. 2 and 3. Fig. 4 represents the increasing nature of anisotropy the fluids spheres.
numerical the strong are given factor for
RELATIVISTIC MODELLING OF STABLE ANISOTROPIC SUPER-DENSE STAR
39
1 0.9 0.8 0.7 0.6 0.5
0.4 0.3 0.2
red shift shiftfor forn=-1 n = −1 red
0.1
red shift shiftfor forn=-3 n = −3 red
0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
1
r/a Fig. 5. Variations of the red-shift (Z) of the energy-momentum tensor for n = −1, c0 a 2 = 0.0829, 10 = 4.2395 and n = −3, c0 a 2 = 0.0028, 10 = 0.7094.
Acknowledgements The authors are very grateful to the Referees for their valuable comments and suggestions, which made the paper better. They are also grateful to the University of Nizwa, Sultanate of Oman, for providing all the necessary facilities and encouragement. REFERENCES [1] L. Herrera, J. Ospino and A. Di Parisco: All static spherically symmetric anisotropic solutions of Einstein’s equations, Phys. Rev. D 77 (2008), 027502. [2] R.C. Tolman: Static solutions of Einstein’s field equations for spheres of fluid, Phys. Rev. 55 (1939), 364. [3] H. A. Buchdahl: General relativistic fluid spheres, Phys. Rev. 116 (1959), 1027. [4] B. V. Ivanov: Gravity, Astrophysics, and String ‘02’, St. Kliment Ohridski University Press, Sofia 2003. [5] R. Ruderman: Pulsars: structure and dynamics, Annu. Rev. Astron. Astrophys. 10 (1972), 427. [6] V. Canuto: Neutron stars: general review, Solvay Conf. Astrophysics Gravitation, Brussels, Sept., 1973. [7] R. Kippenhahm and A. Weigert: Stellar Structure and Evolution, Springer, Berlin 1990. [8] A. I. Sokolov: Phase transitions in a superfluid neutron fluid, J. Exp. Theor. Phys. 79 (1980), 1137. [9] J. Binney and J. S. Tremaine: Galactic Dynamics, Princeton University Press, Princeton, NJ 1987. [10] P. Letelier: Anisotropic fluids with two-perfect-fluid components, Phys. Rev. D 22 (1980), 807. [11] S. S. Bayin: Anisotropic fluid spheres in general relativity, Phys. Rev. D 26 (1982), 1262. [12] R. L. Bowers and E. P. T. Liang: Anisotropic spheres in general relativity, Astrophys. J. 188 (1974), 657. [13] H. Heintzmann and W. Hillebrandt: Neutron stars with an anisotropic equation of state: mass, redshift and stability, Astron. Astrophys. 38, 51 (1975). [14] S. D. Maharaj and R. Maartens: Anisotropic spheres with uniform energy density in general relativity, Gen. Relat. Gravit. 21 (1989), 899. [15] M. K. Gokhroo and A. L. Mehra: Anisotropic spheres with variable energy density in general relativity, Gen. Relat. Gravit. 26 75, (1994). [16] M. S. R. Delgaty and K. Lake: Comput. Phys. Commun. 115, (1998), 395. [17] K. Dev and M. Gleiser: Gen. Relat. Gravit. 34, (2002), 1793.
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