Cmjv01i04p0483 by Journal of Computer and Mathematical Sciences

J. Comp. & Math. Sci. Vol. 1(4), 483-492 (2010).

An Exact Superdense Star Model on Spheroidal Space-Time *M.C. SABU and **RAMESH TIKEKAR *Department of Mathematics, St. Albert’s College, Ernakulam – 682 034 India **Department of Mathematics, Sardar Patel University, Vallabh Vidyanagar, Anand – 388 120 India ABSTRACT Assuming that the physical 3-space t=constant in a superdense star is spheroidal, a class of static spherically symmetric physically viable models based on a particular exact solution of Einstein’s field equations is reported. The models of the class permit densities of the order of 2  10 14 gm cm 3 , radii of the order of a few kilometers and masses up to about the three times the solar masses. Key words: Exact Solutions, Superdense Stars

1. INTRODUCTION

The self interaction of gravitational field as evident from the Einstein field equations of general relativity imposes constraints in obtaining simple exact solutions which may serve as models for relativistic stars. Lack of reliable information about the properties of matter in the central core regions of relativistic compact stars is another hurdle which warrants assumption of general nature. Accordingly it is desired to have some analytic solutions which may serve as easily surveyable models for

these stars. Such closed form solutions1,3,8 will be of astrophysical interest if they comply with certain general basic requirements expected of fluids at ultra high densities and pressure. Vaidya and Tikekar9 have shown that the space-times whose associated physical spaces obtained as t = constant sections have the geometry of a 3-spheroid characterized by the two parameters – k – measuring the oblateness and R spherical nature of the spheroid- are useful in developing, easily surveyable relativistic models for

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M.C. Sabu et al., J.Comp.&Math.Sci. Vol.1(4), 483-492 (2010)

superdense stars such as neutron stars. Knutsen2 has examined the physical plausibility of the class of models of Vaidya and Tikekar and shown that these models are stable with respect to infinitesimal radial pulsations. Yet another class of the models with above geometry has been reported by Tikekar7. Since only a limited number of analytic closed form solutions of Einstein field equations for static spherical distributions of matter can be useful as easily surveyable models for superdense stars, it is necessary to investigate the suitability of the other particular classes of models in this setup. In this paper we have reported a solution for the Einstein field equations describing space-time of a spherical distribution of matter in equilibrium with associated physical 3-space having geometry of a 3-spheroid immersed in a 4-dimensional Euclidean space and examined its suitability for describing the space-time in the interior of a superdense star. In section-2, the general features of the matter distributions in equilibrium in a spheroidal space-time are discussed. A particular closed form exact solution of Einstein’s field equations in this set up reported in the section -3 is used to construct in subsequent sections a relativistic model for a super dense star. Following Vaidya and Tikekar9 and Tikekar7 matter density on the boundary surface of the star is assumed to be of the boundary surface of the star is assumed to be of order 2  1014 gm cm 3 and the esti-

mates for the total mass and size of the star for different values of a parameter  , describing variation of density in the star are obtained. These estimates and other relevant details are given in Table. 1. It is evident that the closed form solution reported here leads to a class of physically viable static models for relativistic stars, the estimates about the mass and size of which have the values in the range corresponding to a neutron star. 2. Matter distribution on spheroidal space-time Following Vaidya and Tikekar9 we begin with the spherical distributions of matter in the form of a perfect fluid at rest with background spacetime described by metric

r2 1 k 2 2 R dr 2  r 2 d 2 ds   r2 1 2 R  r 2 sin 2 d 2  e ( r ) dt 2

(1)

The associated 3-space of this space time (section t = constant) has the geometry of a 3-spheroid immersed in a 4-dimensional Euclidean space. If the physical content of the space-time is in the form of a perfect fluid, with energy momentum tensor

p p  Tij     2 u i u j  2 g ij c  c 

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(2)

M.C. Sabu et al., J.Comp.&Math.Sci. Vol.1(4), 483-492 (2010) where  and p respectively denote matter density and fluid pressure and



u i  0,0,0, e  / 2



(3)

represents unit four-velocity field of matter. Einstein’s field equations

Rij  12 Rg ij  

8G Tij c2

(4)

reduce to the system of three equations

 kr 2  1  2  8G 3(1  k )  3R   2 c2 R2  r2  , 1  k 2  R  

(5)

 r2  1  2  R  8G 1   1  p  2   2 4 2 c r  r ,(6)  r r  1  k 2  R  

485

the spheroidal geometry for the accompanying 3-space of the space-time replaces the usual specification of equation of state of matter. The equation (5) governs the variation of density of matter in the distribution. In7 it is shown that the relativistic condition for hydrostatic equilibrium is

p    2 1 dp c   2 2 c dr r

 4Gp   m(r )  4 r 3   c  2m ( r )  .   1  r  

(8) In this set up, the field equations (7) has the explicit form

 r2  1  k 2  R  1 dp  2 c dr  r2  1  2   R 

and

 r 2  r 2   2    1  2 1  k 2      2 r R   R  

(1  k ) 2(1  k )  r2    0  r   2   1  k R2 R 2  R 2 

(7) Here and in what follows, an overhead prime indicates a differentiation with respect to radial variable r. In this approach the choice of

   4Gpr (1  k ) r   4 r2 2  c  2 R 1  k 2  R  

   p    2  c .     

(9) This is the law determining the pressure gradient linked with the repulsive force counter balancing the gravitational attraction of matter and thus ensuring equilibrium.

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M.C. Sabu et al., J.Comp.&Math.Sci. Vol.1(4), 483-492 (2010)

3. A solution of field equations Equation (7) is a second order, non-linear ordinary differential equation which on adopting new variable  and z defined by

r2   e , z  1 2 R 

(10)

assumes the simple form

solutions of (11), the surveyability of the models constructed using them becomes more and more difficult. Accordingly it is worthwhile to identify tractable models of fluid stars, based on closed form solutions of (7). The equation (11) for particular choice k = -14 is found to admit the general closed form solution

  A1  S 2  2 1  6S 2  3

1  k  kz  ddz  kz ddz  k (k  1)  0. 2

8   8  Bz 1  S 2  S 4  (14) 5   3

(11)

This is a class of linear second order differential equation with class parameter k. The closed form solutions of particular equations of this class can be obtained using suitable methods for their integration. Wils6 has couched equation of type (7) into the form of Riccati equation and studied physical relevance of solutions obtained for the two particular choices of the respective class parameter. Using the transformation, for k < 0

k 2 z k 1

u2 

(12)

the differential equation (11) reduces to the form 2

1  u  ddu  u ddu  (1  k )  0 2

(13)

which was studied by Vaidya and Tikekar9 for the value of k = -2. A general method for obtaining closed form solutions of the equation7 is extensively discussed by Maharaj and Leach4. With increasing complexity of the nature of

where S 2 

14 2 z , A and B denoting 15

arbitrary constants of integration. The space-time metric for this solution written out explicitly will be

ds 2  15

1 S 2 2 dr  r 2 d 2 2 z



  sin 2 d 2   A 1  S 2 





 1  6S   3

8   8  Bz 1  S 2  S 4   dt 2 . (15) 5   3 The matter density and fluid pressure of the accompanying fluid distribution are given by

 14r 2  1  2  8G 45  3R    2 c2 R2  r2  1  14 2  R  

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487

  56 2 40 4  2 12 2 4  15 A(1  S ) (1  9 S  10S )  17 Bz1  17 S  17 S  8G   p  4 3 c  8  . (17)  8 15R 2 (1  S 2 )  A 1  S 2 2 1  6 S 2  Bz 1  S 2  S 4  5   3 







In the subsequent sections we have discussed the physical plausibility of the star models based on this solution.

From the expression for density it can

4. Physical Plausibility

ghout the distribution. Further following the arguments in7, it is observed that the equation for hydrostatic equilibrium (11) implies that at all point r < R, if the pressure p(r) > 0, the pressure

Since the solution obtained is independent of any assumption about the inter particle interaction, it is necessary to check its physical plausibility. For this, one should examine the following requirements expected to fulfill in this domain of validity: (i) The matter density  and fluid pressure p should be positive everywhere.

d dp (ii) The gradients and should dr dr be negative. (iii) The speed of sound should never exceed the speed of light in the distribution and in fact should be less than it. (iv) The metric (15) should join continuously with the exterior Schwarzschild metric 1

 2m  2 2 2 ds  1   dr  r d r   2



be seen that  > 0 and

gradient

d <0 throudr

d <0. The density and dr

pressure attain the values and p0 given by

8G 45 0  2 2 c R

(19)

 



8G 25 59 15 A  19 B p0  2 4 c R 23 15 A  107 B



(20)

at the centre r = 0 of the distribution. The positivity of the pressure at the centre is ensured by imposing the condition on the constants of integration as -0.83 < B/A < 12.02

(21)

If we further impose the condition

 2m  2  sin 2 d 2  1   dt . (18) r  



0  3

p0  0 , such that the strong c2

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M.C. Sabu et al., J.Comp.&Math.Sci. Vol.1(4), 483-492 (2010)

energy condition is satisfied at the centre gives more restricted condition on A and B such that 2.1  B/A < 12.02

(22)

At the surface r = a, the pressure must be zero. This condition together with the continuity of metric coefficient gives a

4G m  2   2  ( ) d c 0

m

15 2

a3  a2 R 1  14 2 R  2

  

(23)

 2m  2 1    A 1 S a a  

  Bz

We introduce the density variation parameter

 14 a 2  1  3 2  R  a    2 0  a2  1  14 2  R  

(2 6)

where  a and  0 respectively repreesent matter density at the boundary and at the centre. If  0 and  have e specified values, equations (19) and

and 1

The conditions (24) and (25) determines the constants A and B while the condition (23) determines the total mass m of the distribution.



 1  6S  3

8 4   8 2 1  S a  S a  5  3 

(24)

(26) respectively determinesand R 2 2 and a

hence a and R. The equation

(23) subsequently determines the m = the mass content of the fluid sphere.

The pressure p = 0 at r = a. This gives 2

 15 A(1  S a ) (1  9S 2

 10S a )

40 4   56  17 Bz a 1  S 2 a  S a  (25) 17  17 

Following the approach developed in (7) the matter content of the spacetime of (15) we get the expression for

dp for k = -14 as d

p  8G r2  4   GR    2 15  4 pR 1  14 2 c  c R  dp   d  r 2  r2     1051  2  5  14 2  R   R  2

  r2   1  14 2   R   

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M.C. Sabu et al., J.Comp.&Math.Sci. Vol.1(4), 483-492 (2010) If the matter distribution complies with the strong energy condition at the centre i.e.,  0  3 p 0  0 , (27) implies (28)

At the boundary r = a,

 a 2  14a 2  1  14 2 1   R  3R 2  2 45   c 56  a 2  a2  1  2  5  14 2  R   R  (29)

which will be less than c2, if a R is less than 0.46. It is observed that this requirement is fulfilled for the models with   0.3 , satisfying strong energy condition. Since

guration. Subsequently speed of sound will not exceed the speed of light through out the space-time. 5. DISCUSSION

 dp     0.43c 2 . d    r 0

 dp     d  r  a

489

dp d

does not vanish

When the thermonuclear sources of energy in the interior of a star are exhausted, it begins to contract under the influence of gravitational interaction of its matter content until it ends up in its final fate as a white dwarf, neutron star or a black hole. The models proposed by Vaidya and Tikekar 9 describe a superdense star formed during these last stages of stellar evolution with densities of their matter content in the range of 1014 – 1016 gm cm-3. We assume that at the boundary r = a, the density of the star is  a  2  1014 gm cm 3 which corresponds to that of neutron star. Then, adopting the scheme explained in section–3 giving different values of  we calculate R, a, m, B and A which are given in Table 1.

anywhere inside the spherical region dp  c 2 throughout the confir  a, d Table 1 Sr. No.  R (Km) a (Km) a/ R M/ M0 1 0.90 104.29 7.14 0.07 0.16 2 0.80 98.34 9.99 0.10 0.46 3 0.70 91.98 12.11 0.13 0.86 4 0.60 85.16 13.80 0.16 1.35 5 0.50 77.74 15.20 0.20 1.92 6 0.40 69.53 16.34 0.24 2.59 7 0.30 60.21 17.23 0.29 3.34 8 0.20 49.16 17.83 0.36 4.19 9 0.10 34.76 17.97 0.52 5.15 Note: M = mc2/G, M0= Mass of the sun

B 9.22 8.54 7.78 6.92 5.91 4.71 3.20 1.19 -1.50

A 0.91 1.00 1.10 1.19 1.28 1.35 1.38 1.30 0.82

B/A 10.11 8.48 7.06 5.79 4.62 3.48 2.31 0.91 -1.82

Table. 1: Masses and equilibrium radii corresponding to  a  2  1014 gm cm 3 , for the class of relativistic star models. Journal of Computer and Mathematical Sciences Vol. 1 Issue 4, 30 June, 2010 Pages (403-527)

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From Table 1, it can be seen that mass and radius of the configuration are decreasing functions of density variation parameter. The equilibrium radius and the total mass of each of these models is of the order of the radius and mass of a neutron star. It has been verified using numerical methods that all the models corresponding to the values of   0.3 comply with the physical requirements   0, p  0,  

3p 0 c2

are fulfilled through out the distribution. Astrophysists claim5 that they

Figure. I. Graphs of ˆ 

know the equation of the state for neutron material if the density is less than a certain fiduacial density  f  5.1 

1014 gm cm 3 . For the models with

  0.3,  0   f .

This make the

model with   0.3 interesting. For the model with   0.3 , the

8Gp 8G 8Gp , 4 and 3  4 2 c c c with radial variable throughout the distribution are shown graphically in Figure. I. variations of

8G 8G  , pˆ  4 p and 3 pˆ against radius 2 c c

The variation of pressure with density for the same model is also shown graphically in Figure. II. Journal of Computer and Mathematical Sciences Vol. 1 Issue 4, 30 June, 2010 Pages (403-527)

M.C. Sabu et al., J.Comp.&Math.Sci. Vol.1(4), 483-492 (2010)

Figure. II. Graphs of ˆ 

The maximum mass of the configuration with   0.3 is obtained at the radius 17.2 kms. If the condition



3p p  0 is relaxed to   2  0 , 2 c c

models exceeding this mass at radii higher than 17.2 kms are admissible provided the requirement (21) is fulfilled. Thus the space-time metric (15) gives us a series of equilibrium configurations each having surface density,

491

8G 8G ˆ  against p  p c2 c4 mass and radii of the same order as in a neutron star. The star models of this class are smaller in size and mass than the corresponding models for k = -2 and k =-7 with the same values for density variation parameter . The models with k = -14 admit more density variation. REFERENCES 1. Buchdal, H. A., Phys. Rev., General Relativistic Fluid Spheres, 1027 – 1034, 116 (1959).

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M.C. Sabu et al., J.Comp.&Math.Sci. Vol.1(4), 483-492 (2010)

2. Knutsen, H., Mon. Not. R. Astron. Soc., On stability and Physical Properties of an Exact Relativistic model for a dense star, 163-174232 (1988). 3. Leibovitz, C., Phys. Rev., Spherically Symmetric Static Solutions of Einstein’s Equations, 1664 – 1670, 185, (1969). 4. Maharaj, S. D., and Leach, P.G.L., J. Math. Phys., 37, Exact Solutions for the Tikekar Superdense Star, 430438, (1996). 5. Mukherjee, S., Paul, B.C. and Dadhich, N., Class. Quantum. Grav., General Solution for a Relativistic Spheroidal Star, 3475-3480, 14, (1997).

6. Patric Wils., Gen. Rela. Grav., Physical Properties of Buchadahl’s ThreeParameter Static Spherically Symmetric Perfect Fluid Metrics, 539 – 552, 5, (1990). 7. Tikekar, R., J. Math. Phys., Exact model for a Relativistic Star, 2454 -2458, 31, (1990). 8. Tolman, R.C., Phys. Rev., Static Solution of Einstein’s Field Equations for spheres of fluids, 364– 373, 55, (1939). 9. Vaidya, P.C. and Tikekar R., J. Astrophys. Astron., Exact Relativistic Model for a Superdense Star, 325 – 334, 3, (1982).

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