A new class of solutions of anisotropic charged distributions on pseudo-spheroidal spacetime

A new class of solutions of anisotropic charged distributions on pseudo-spheroidal spacetime

B. S. Ratanpal11affiliationmark: Department of Applied Mathematics, Faculty of Technology & Engineering, The M. S. University of Baroda, Vadodara - 390 001, India V. O. Thomas22affiliationmark: Department of Mathematics, Faculty of Science, The M. S. University of Baroda,
Vadodara - 390 002, India
D. M. Pandya33affiliationmark: Department of Mathematics & Computer Science, Pandit Deendayal Petroleum University, Raisan, Gandhinagar - 382 007, India
Abstract

In the present article a new class of exact solutions of Einstein’s field equations for charged anisotropic distribution is obtained on the background of pseudo-spheroidal spacetime characterized by the metric potential , where and are geometric parameters of the spacetime. The radial pressure and electric field intensity are taken in the form and . The bounds of geometric parameter and the parameter appearing in the expression of are obtained by imposing the requirements for a physically acceptable model. It is found that the model is in good agreement with the observational data of number of compact stars like 4U 1820-30, PSR J1903+327, 4U 1608-52, Vela X-1, PSR J1614-2230, Cen X-3 given by Gangopadhyay et al [Gangopadhyay T., Ray S., Li X-D., Dey J. and Dey M., Mon. Not. R. Astron. Soc. 431 (2013) 3216]. When the model reduces to the uncharged anisotropic distribution given by Ratanpal et al. [Ratanpal B. S., Thomas V. O. and Pandya D. M., arXiv:1506.08512 [gr-qc](2015)]

General relativity; Exact solutions; Anisotropy; Relativistic compact stars; Charged distribution

1 Introduction

Mathematical model for generating superdense compact star models compatible with observational data has got wide attention among researchers. A number of papers have been appeared in literature in the recent past along this direction considering matter distribution incorporating charge (Maurya and Gupta, 2011a, b, c; Pant and Maurya, 2012; Maurya et al, 2015). It has been suggested, as a result of theoretical investigations of Ruderman (1972) and Canuto (1974), that matter may not be isotropic in high density regime of . Hence it is pertinent to construct charged distribution incorporating anisotropy in pressure. Bonner (1960, 1965) has shown that a spherical distribution of matter can retain its equilibrium by counter balancing the gravitational force of attraction by Coulombian force of repulsion due to the presence of charge. It was shown by Stettner (1973) that a spherical distribution of uniform density accompanied by charge is more stable than distribution without charge. The study of charge distributions on spheroidal spacetimes have been carried out by Patel and Kopper (1987), Tikekar and Singh (1998), Sharma et al (2001), Gupta and Kumar (2005), Komathiraj and Maharaj (2007). The spheroidal spacetime is found to accommodate superdense stars like neutron stars in both charged and uncharged cases. Study of strange stars and quark stars in the presence of electric charge have been done by Sharma et al (2006), Sharma and Mukherjee (2001), Sharma and Mukherjee (2002). Recently charged fluid models have also been studied by Maurya and Gupta (2011a, b, c), Pant and Maurya (2012) & Maurya et al (2015).

In this paper, we have obtained a new class of solutions for charged fluid distribution on the background of pseudo spheroidal spacetime. Particular choices for radial pressure and electric field intensity are taken so that the physical requirements and regularity conditions are not violated. The bounds for the geometric parameter and the parameter associated with charge, are determined using various physical requirements that are expected to satisfy in its region of validity. It is found that these models can accommodate a number of pulsars like 4U 1820-30, PSR J1903+327, 4U 1608-52, Vela X-1, PSR J1614-2230, Cen X-3, given by Gangopadhyay et al (2013). When the model reduces to the uncharged anisotropic distribution given by Ratanpal et al (2015).

In section 2, we have solved the field equations and in section 3, we have obtained the bounds for different parameters using physical acceptability and regularity conditions. In section 4, We have displayed a variety of pulsars in agreement wit the charged pseudo-spheroidal model developed. In particular we have studied a model for various physical conditions throughout the distribution and discussed the main results at the end of this section.

2 Spacetime Metric

We shall take the interior spacetime metric representing charged anisotropic matter distribution as

 ds2=eν(r)dt2−⎛⎜⎝1+Kr2R21+r2R2⎞⎟⎠dr2−r2(dθ2+sin2θdϕ2), (1)

where and are geometric parameters and . This spacetime, known as pseudo-spheroidal spacetime, has been studied by number of researchers Tikekar and Thomas (1998, 1999, 2005); Thomas et al (2005); Thomas and Ratanpal (2007); Paul et al (2011); Paul and Chattopadhyay (2010); Chattopadhyay and Paul (2012) have found that it can accommodate compact superdense stars.

Since the metric potential is chosen apriori, the other metric potential is to be determined by solving the Einstein-Maxwell field equations

 Rji−12Rδji=8π(Tji+πji+Eji), (2)

where,

 Tji=(ρ+p)uiuj−pδji, (3)
 πji=√3S[cicj−12(uiuj−δji)], (4)

and

 Eji=14π(−FikFjk+14FmnFmnδji). (5)

Here , , , and , respectively, denote the proper density, fluid pressure, unit-four velocity, magnitude of anisotropic tensor and a radial vector given by . denotes the anti-symmetric electromagnetic field strength tensor defined by

 Fij=∂Aj∂xi−∂Ai∂xj, (6)

which satisfies the Maxwell equations

 Fij,k+Fjk,i+Fki,j=0, (7)

and

 ∂∂xk(Fik√−g)=4π√−gJi, (8)

where denotes the determinant of , is four-potential and

 Ji=σui, (9)

is the four-current vector where denotes the charge density.

The only non-vanishing components of is . Here

 F01=−eν+λ2r2∫r04πr2σeλ/2dr, (10)

and the total charge inside a radius is given by

 q(r)=4π∫r0σr2eλ/2dr. (11)

The electric field intensity can be obtained from , which subsequently reduces to

 E=q(r)r2. (12)

The field equations given by (2) are now equivalent to the following set of the non-linear ODE’s

 1−e−λr2+e−λλ′r=8πρ+E2, (13)
 e−λ−1r2+e−λν′r=8πpr−E2, (14)
 e−λ(ν′′2+ν′24−ν′λ′4+ν′−λ′2r)=8πp⊥+E2, (15)

where we have taken

 pr=p+2S√3, (16)
 p⊥=p−S√3. (17)

Because , the metric potential is known function of . The set of equations (13) - (15) are to be solved for five unknowns , , , and . So we have two free variables for which suitable assumption can be made. We shall assume the following expressions for and .

 8πpr=K−1R21−r2R2(1+Kr2R2)2, (18)
 E2=α(K−1)R2r2R2(1+Kr2R2). (19)

It can be noticed from equation (18) that vanishes at and hence we take the geometric parameter as the radius of distribution. Further for all values of in the range . It can also be noted that is regular at . On substituting the values of and in (14) we obtain, after a lengthy calculation

 eν=CR[K2−(2+α)K+α+1]K(1+Kr2R2)(K+α+12K)(1+r2R2)K−α−32, (20)

where is a constant of integration. Hence, the spacetime metric takes the explicit form

 ds2 = CR[K2−(2+α)K+α+1]K(1+Kr2R2)(K+α+12K)(1+r2R2)K−α−32dt2

The constant of integration can be evaluated by matching the interior spacetime metric with Riessner-Nordström metric

 ds2=(1−2mr+q2r2)dt2−(1−2mr+q2r2)−1dr2−r2(dθ2+sin2θdϕ2), (22)

across the boundary . This gives

 M=R2[K2+α(K−1)−1](1+K)2, (23)

and

 C=R−[K2−(2+α)K+α+1]K(1+K)−(3K+α+12K)2(α−K+52). (24)

Here denotes the total mass of the charged anisotropic distribution.

3 Physical Requirements and Bounds for Parameters

The gradient of radial pressure is obtained from equation (18) in the form

 8πdprdr=−2r(K−1)R41+2K−Kr2R2(1+Kr2R2)3<0. (25)

It can be noticed from equation (25) that the radial pressure is decreasing function of . Now, equation (13) gives the density of the distribution as

 8πρ=(K−1R2)3+(K−α)r2R2(1+Kr2R2)2. (26)

The conditon is clearly satisfied and gives the following inequality connecting and .

 0≤α<3+K. (27)

Differentiating (26) with respect to , we get

 8πdρdr=−2r(K−1)R45K+α+K(K−α)r2R2(1+Kr2R2)3. (28)

It is observed that and leads to the inequality

 K2−K(α−5)+α≥0. (29)

The inequality (29) together with the condition give a bound for as

 0≤α

The expression for is

 8πp⊥=4K−4+X1r2R2+X2r4R4+X3r6R6R2(4+Y1r2R2+Y2r4R4+Y3r6R6+4K3r8R8), (31)

where, , , , , and .

The condition at the boundary imposes a restriction on and respectively given by

 K>2√3−1 (32)

and

 0≤α<10+5K+K2K−1−√89+102K+57K2+8K3(K−1)2. (33)

The expression for is given by

 dp⊥dr=−r(8K2+(12α+8)K−12α−16+A1r2R2+A2r4R4+A3r6R6+A4r8R8)R4(2+B1r2R2+B2r4R4+B3r6R6+B4r8R8+B5r10R10+2K4r12R12), (34)

where, , , , , , , , and .

The value of at the origin and gives the following bounds for and respectively

 2√13−5

and

 0≤α

In order to examine the strong energy condition, we evaluate the expression at the centre and on the boundary of the star. It is found that

 (ρ−pr−2p⊥)(r=0)=0, (37)

and gives the bound on and , namely

 1
 0≤α<8+3K+K2K−1+√41+46K+49K2+8K3(K−1)2. (39)

The expressions for adiabatic sound speed and in the radial and transverse directions, respectively, are given by

 dprdρ=1+2K−Kr2R25k+α+K(K−α)r2R2, (40)

and

 dp⊥dρ=(1+Kr2R2)3[8K2+(12α+8)K−12α−16+C1r2R2+C2r4R4+C3r6R6+C4r8R8]2(K−1)[5K+α+K(K−α)r2R2][2+D1r2R2+D2r4R4+D3r6R6+D4r8R8+D5r10R10+2K4r12R12], (41)

where, , , , , , , , and .

The condition is evidently satisfied at the centre whereas at the boundary it gives a restriction on as

 0≤α1. (42)

Further at the centre will lead to the following inequalities

 K>43 (43)

and

 0≤α<12(3K−4). (44)

Moreover at the boundary , we have the following restrictions on and .

 −5+2√13≤K<5 (45)

and

 0≤α≤K3+10K2+25K−20K2−6K+5+√16K5+233K4+252K3+278K2−788K+265(K2−6K+5)2, (46)

The necessary condition for the model to represent a stable relativistic star is that throughout the star. at gives a bound on which is identical to (27). Further, as and hence the condition is automatically satisfied. It can be noticed that at , showing the regularity of the charged distribution.

The upper limits of in the inequalities (27), (30), (33), (36), (39), (42) and (44) for different permissible values of are shown in Table 1. It can be noticed that for the bound for is

4 Application to Compact Stars and Discussion

In order to compare the charged anisotropic model on pseudo-spheroidal spacetime with observational data, we have considered the pulsar PSR J1614-2230 whose estimated mass and radius are and . On substituting these values in equation (23) we have obtained the values of adjustable parameters and as and respectively which are well inside their permitted limits. Similarly assuming the estimated masses and radii of some well known pulsars like 4U 1820-30, PSR J1903+327, 4U 1608-52, Vela X-1, PSR J1614-2230, Cen X-3, we have displayed the values of the parameters and , the central density , surface density , the compactification factor , and charge inside the star in Table 2. From the table it is clear that our model is in good agreement with the most recent observational data of pulsars given by Gangopadhyay et al (2013).

In order to examine the nature of physical quantities throughout the distribution, we have considered a particular star PSR J1614-2230, whose tabulated mass and radius are . Choosing and , we have shown the variations of density and pressures in both the charged and uncharged cases in Figure 1, Figure 2 and Figure 3. It can be noticed that the pressure is decreasing radially outward. The density in the uncharged case is always greater than the density in the charged case. Similarly the radial pressure and transverse pressure are decreasing radially outward. Similar to that of density, and in the uncharged case accommodate more values compared to charged case.

The variation of anisotropy shown in Figure 4 is initially decreasing with negative values reaches a minimum and then increases. In this case also anisotropy takes lesser values in the charged case compared to uncharged case. The square of sound in the radial and transverse direction (i.e. and ) are shown in Figure 5 and Figure 6 respectively and found that they are less than 1. The graph of against radius is plotted Figure 7. It can be observed that it is non-negative for and hence strong energy condition is satisfied throughout the star.

A necessary condition for the exact solution to represent stable relativistic star is that the relativistic adiabatic index given by should be greater than The variation of adiabatic index throughout the star is shown in Figure 8 and it is found that throughout the distribution both in charged and uncharged case. Though we have not assumed any equation of state in the explicit form and , we have shown the relation between against in the graphical form as displayed in Figure 9 and Figure 10. For a physically acceptable relativistic star the gravitational redshift must be positive and finite at the centre and on the boundary. Further it should be a decreasing function of . Figure 11 shows that this is indeed the case. Finally we have plotted the graph of against which is displayed in Figure 12. Initially increases from and reaches a maximum values and then decreases radially outward. The model reduces to the uncharged anisotropic distribution given by Ratanpal et al (2015) when

Acknowledgement

The authors would like to thank IUCAA, Pune for the facilities and hospitality provided to them for carrying out this work.

References

• Paul and Chattopadhyay (2010) Chattopadhyay P. C. and Paul B. C., Pramana- J. of phys. 74 (2010) 513.
• Tikekar and Jotania (2005) Tikekar R. and Jotania K., Int. J. Mod. Phys. D 14 (2005) 1037.
• Tikekar and Jotania (2007) Tikekar R. and Jotania K., Pramana- J. of phys. 68 (2007) 397.
• Tikekar (1990) Tikekar R. , J. Math. Phys. 31, (1990) 2454.
• Ruderman (1972) Ruderman R., Astro. Astrophys. 10 (1972) 427.
• Canuto (1974) Canuto V., Annu. Rev. Astron. Astrophys. 12 (1974) 167.
• Tikekar and Thomas (1998) Tikekar R. and Thomas V. O., Pramana- J. of phys. 50 (1998) 95.
• Tikekar and Thomas (2005) Tikekar R. and Thomas V. O., Pramana- J. of phys. 64 (2005) 5.
• Thomas et al (2005) Thomas V. O., Ratanpal B. S. and Vinodkumar P. C., Int. J. Mod. Phys. D 14 (2005) 85.
• Thomas and Ratanpal (2007) Thomas V. O. and Ratanpal B. S., Int. J. Mod. Phys. D 16 (2007) 9.
• Komathiraj and Maharaj (2007) Komathiraj K. and Maharaj S. D., Intenational Journal of Modern Physics D 16 (2007) 1803.
• Paul et al (2011) Paul B. C., Chattopadhyay P. K., Karmakar S. and Tikekar R., Mod. Phys. Lett. A 26 (2011) 575.
• Sharma and Ratanpal (2013) Sharma R. and Ratanpal B. S., Int. J. Mod. Phys. D 13 (2013) 1350074.
• Gangopadhyay et al (2013) Gangopadhyay T., Ray S., Li X-D., Dey J. and Dey M., Mon. Not. R. Astron. Soc. 431 (2013) 3216.
• Tikekar and Thomas (1999) Tikekar R. and Thomas V. O., Pramana J. Phys. 52 (1999) 237.
• Chattopadhyay and Paul (2012) Chattopadhyay P. C., Deb R. and Paul B. C., Intenational Journal of Modern Physics D 21 (2012) 1250071.
• Bonner (1960) Bonner W. B., J. Phys. 160 (1960) 59.
• Bonner (1965) Bonner W. B., Mon. Not. R. Astron. Soc. 29 (1965) 443.
• Stettner (1973) Stettner R., Ann.Phys. 80 (1973) 212.
• Patel and Kopper (1987) Patel. L. K. and Kopper, Austr. J. Phys. 40 (1987) 441.
• Tikekar and Singh (1998) Tikekar R. and Singh G. P., Gravitation and Cosmology 4 (1998) 294.
• Sharma et al (2001) Sharma R., Mukherjee S. and Maharaj S. D., Gen. Relativ. Gravit. 33 (2001) 999.
• Gupta and Kumar (2005) Gupta Y. K. and Kumar N., Gen. Relativ. Gravit. 37 (2005) 575.
• Sharma et al (2006) Sharma R., Karmakar S. and Mukherjee S., Intenational Journal of Modern Physics D 15 (2006) 405.
• Sharma and Mukherjee (2001) Sharma R. and Mukherjee S., Modern Physics Letters A 16 (2001) 1049.
• Sharma and Mukherjee (2002) Sharma R. and Mukherjee S., Modern Physics Letters A 17 (2002) 2535.
• Maurya and Gupta (2011a) Maurya S. K. and Gupta Y. K., Astrophys. Space Sci. 331 (2011a) 135.
• Maurya and Gupta (2011b) Maurya S. K. and Gupta Y. K., Astrophys. Space Sci. 332 (2011b) 155.
• Maurya and Gupta (2011c) Maurya S. K. and Gupta Y. K., Astrophys. Space Sci. 333 (2011c) 415.
• Pant and Maurya (2012) Pant N. and Maurya S. K., App. Math. Comp. 218 (2012) 8260.
• Maurya et al (2015) Maurya S. K. and Gupta Y. K. and Ray S., arXiv:1502.01915 [gr-qc] (2015).
• Ratanpal et al (2015) Ratanpal B. S., Thomas V. O. and Pandya D. M., arXiv:1502.01915 [gr-qc] (2015).
You are adding the first comment!
How to quickly get a good reply:
• Give credit where it’s due by listing out the positive aspects of a paper before getting into which changes should be made.
• Be specific in your critique, and provide supporting evidence with appropriate references to substantiate general statements.
• Your comment should inspire ideas to flow and help the author improves the paper.

The better we are at sharing our knowledge with each other, the faster we move forward.
The feedback must be of minimum 40 characters and the title a minimum of 5 characters