Varieties of Parametric Classes of Exact Solutions in General Relativity Representing Static Fluid Balls

Abstract

We have presented a method of obtaining parametric classes of spherically symmetric analytic solutions of the general relativistic field equations in canonical coordinates. A number of previously known classes of solutions have been rediscovered which describe perfect fluid balls with infinite central pressure and infinite central density though their ratio is positively finite and less than one. From the solution of one of the newly discovered classes, we have constructed a causal model in which outmarch of pressure and density is positive and monotonically decreasing, and pressure-density ratio is positive and less than one throughout within the balls. Corresponding to this model, we have maximized the Neutron star mass 2.40𝑀_Θ with the linear dimensions of 28.43 kms and surface red shift of 0.4142.

1. Introduction

Due to nonlinearity of the Einstein’s field equations, many attempts have been reported to obtain parametric classes of exact solutions of representing perfect fluid ball in equilibrium such as Tolman [1], Wyman [2], Kuchowich [3], Pant and Sah [4], D.N. Pant and N. Pant [5], Pant [6, 7], Pant [8], Delgaty, Lake [9], Lake [10], Tewari and Pant [11], and Maurya, Gupta [12]. These solutions have four arbitrary constants. The usual boundary conditions determine three arbitrary constants leaving one undetermined; such a solution represents a class of solutions, with undetermined constant being a parameter. The edge of a parametric class of solutions over an ordinary solution lies in the choice of associated parameter which provides us various models of relativistic stars with realistic equation of state. Moreover, by imposing realistic conditions on parameter, one may get physical and significant solutions. Thus in the light of importance of parametric class, in this paper, we present a variety of parametric classes of solutions.

2. Field Equations and Method of Obtaining Parametric Classes of Solutions

We consider the static and spherically symmetric metric in canonical coordinates𝑑𝑠²=−𝑒^𝜆𝑑𝑟²−𝑟²𝑑𝜃²+sin²𝜃𝑑𝜙²+𝑒^𝜐𝑑𝑡²,(1) where 𝜆 and 𝜐 are field variables and functions of 𝑟 only. The field equations of gravitation for a nonempty space-time are𝑅_𝑖𝑗−12𝑅𝑔^𝑖𝑗=−8𝜋𝐺𝑐⁴𝑇_𝑖𝑗,(2) where 𝑅_𝑖𝑗 is a Ricci tensor, 𝑇_𝑖𝑗 is energy-momentum tensor, and 𝑅 the scalar curvature. The energy-momentum tensor 𝑇_𝑖𝑗 is defined as𝑇_𝑖𝑗=𝑝+𝜌𝑐²𝑣_𝑖𝑣_𝑗−𝑝𝑔_𝑖𝑗,(3) where 𝑝 denotes the pressure distribution, 𝜌 the density distribution, and 𝑣_𝑖 the velocity vector, satisfying the relation𝑔_𝑖𝑗𝑣^𝑖𝑣^𝑗=1.(4)

Since the field is static, therefore, only a nonzero component of velocity is 𝑣⁴𝑣⁴=1√𝑔₄₄.(5)

Thus, under these conditions, the field equations of general relativity for a perfect fluid ball the physical variables 𝑝(𝑟) and 𝜌(𝑟) are (Tolman[1])8𝜋𝐺𝑐⁴𝑝=𝑒^−𝜆𝜐𝑟^/+𝑟1²−𝑟1²8𝜋𝐺,(6)𝑐²𝜌=𝑒^−𝜆𝜆𝑟^/−𝑟1²+𝑟1²𝑑,(7)𝑑𝑟𝑒^−𝜆−1𝑟²+𝑑𝑑𝑟𝑒^−𝜆𝜐^/2𝑟+𝑒^{−𝜆−𝜐}𝑑𝑑𝑟𝑒^𝜐𝜐^/2𝑟=0,(8) where prime (/) denotes differentiation with respect to 𝑟. The problem consists of solving (8) for 𝜆 and 𝜐 and obtaining 𝑝(𝑟) and 𝜌(𝑟) from (6) and (7).

By using the transformation𝑈=𝑟^𝑚−1𝑒^𝑚𝜐/2,𝑉=𝑒^−𝜆.(9) Equation (8) reduces to the following linear differential equation in 𝑉:𝑑𝑉𝑑𝑟−2𝑑𝑑𝑟log𝑟^4−1/𝑚𝑈^1−1/𝑚=−2𝑚𝑈𝑟𝑈^/+𝑈−𝑟𝑟2𝑈𝑚^/𝑈+𝑈𝑉𝑟𝑟𝑈^/+𝑈.(10) On solving (10), we get𝑉=𝑒^−𝜆=𝑟^8−2/𝑚𝑈^{2(1−1/𝑚)}𝑟𝑈^/+𝑈²𝐴−2𝑚𝑟𝑈^/+𝑈𝑈^−1+2/𝑚𝑒^{∫(4𝑚𝑈𝑑𝑟/𝑟(𝑟𝑈/+𝑈))}𝑟^9−2/𝑚𝑑𝑟𝑒^{−∫(4𝑚𝑈𝑑𝑟/𝑟(𝑟𝑈/+𝑈))},(11)where “𝐴” is an arbitrary constant. Our aim is to explore the possibilities of choosing 𝑈 such that the right-hand side of (11) becomes integrable. In this paper, we assume𝑒^{∫(4𝑚𝑈𝑑𝑟/𝑟(𝑟𝑈/+𝑈))}=𝑟^𝑙𝑟𝑈^/+𝑈^𝑛;(12)𝑙, 𝑚, and 𝑛 are arbitrary constants. Equation (12) results into a second degree homogenous differential equation in U𝑛𝑟²𝑈^//+(𝑙+2𝑛)𝑟𝑈^/+(𝑙−4𝑚)𝑈=0.(13) The solution is𝑈=𝐶₁𝑟^{𝑎+𝑏−1}+𝐶₂𝑟^{𝑎−𝑏−1},(14) where𝑎=𝑛−𝑙2𝑛,𝑏=21𝑛(𝑛−𝑙)²+16𝑚𝑛,(15) provided, 𝑛≠0.

𝐶₁ and 𝐶₂ are arbitrary constants. Also (11) is simplified into𝑉=𝑒^−𝜆=𝑟^{8+𝑛−𝑙−(𝑎−𝑏)(𝑛+2/𝑚)}[𝐴−2𝐼](𝑎+𝑏)𝐶₁𝑟^2𝑏+(𝑎−𝑏)𝐶₂^𝑛+2𝐶₁𝑟^2𝑏+𝐶₂^{2(−1+1/𝑚)},(16) where 𝐼≡𝑚𝑟^{𝑙−𝑛−9+(𝑎−𝑏)(𝑛+2/𝑚)}(𝑎+𝑏)𝐶₁𝑟^2𝑏+(𝑎−𝑏)𝐶₂^𝑛+1×𝐶₁𝑟^2𝑏+𝐶₂^−1+2/𝑚𝑑𝑟.(17) The solution is complete if (17) is integrated. In the foregoing sections, we shall discuss the method of solving (17), which will result into the variety of classes of solutions.

3. Varieties of Classes of Solutions

To explore the integrability of (17), we substitute𝑙−𝑛−9+(𝑎−𝑟𝑏)^2𝑏=𝑛+𝑦,2(18)𝑚=2𝑏𝛼−1.(19)

So that (17) transforms into𝐼≡𝑚2𝑏𝑦^𝛼−1(𝑎+𝑏)𝐶₁𝑦+(𝑎−𝑏)𝐶₂^𝑛+1×𝐶₁𝑦+𝐶₂^−1+2/𝑚𝑑𝑦.(20) It is easy to see that right-hand side of (20) can be integrated by parts with the restriction on the exponents of integrand to be nonnegative integral values. Thus we arrive at varieties of classes of solutions.

3.1. Class I (new class of solutions)

We assume 𝑛+1=0 and −1+2/𝑚=0. In view of (18) and (20), the resulting class of solutions is 𝑒_−𝜆𝑒^𝜐=𝐶₁𝑟^{𝑎+𝑏−2}+𝐶₂𝑟^{𝑎−𝑏−2},=𝐴𝑟^7−𝑙−4/(𝑙−7)𝐶₁𝑟^2𝑏+𝐶₂(𝑎+𝑏)𝐶₁𝑟^2𝑏+(𝑎−𝑏)𝐶₂,(21) where𝑎=1+𝑙2,𝑏=√(1+𝑙)²−32−2.(22) We observe that (𝑒^𝜐)_𝑟=0 becomes singular for all values except √32−1≤𝑙≤5. It may be pointed out here that for 𝑙=5, we rediscover Tolman’s IV solution which is the only member of the class giving rise to finite central pressure and central density.

4. Properties of the New Class of Solutions

In view of (21), we obtain from (6) and (7), the pressure and density distribution, respectively,^{8𝜋𝐺𝑐41𝑝=𝑟2𝐴𝑟7−𝑙−4/𝑙−7(𝑎+𝑏−1)𝐶1𝑟2𝑏+(𝑎−𝑏−1)𝐶2(𝑎+𝑏)𝐶1𝑟2𝑏+(𝑎−𝑏)𝐶2,−18𝜋𝐺}_{𝑐21𝜌=𝑟2𝐴𝑟7−𝑙−4/𝑙−74𝑏2𝐶1𝐶2𝑟2𝑏+(𝑙−8)𝐴𝑟7−𝑙×𝐶+4/𝑙−71𝑟2𝑏+𝐶2(𝑎+𝑏)𝐶1𝑟2𝑏+(𝑎−𝑏)𝐶2(𝑎+𝑏)𝐶1𝑟2𝑏+(𝑎−𝑏)𝐶22.+1}(23)

The central values of pressure and density are infinite; however, the limiting value of their ratio is finite and equal to the limiting value of 𝑑𝑝/𝑑𝜌:𝜌𝑝𝑐²_𝑟→0=1𝑐²𝑑𝑑𝜌𝑝_𝑟→0=−[4(𝑎+𝑏−1)+(𝑙−7)(𝑎+𝑏)][4+(𝑙−7)(𝑎+𝑏)].(24)

The causality condition at the centre is valid for all those values of 𝑙 satisfying the inequality √32−1≤𝑙≤5.

In addition to the parameter 𝑙, the solutions (21) contain three arbitrary constants 𝐶₁, 𝐶₂, and 𝐴. These are to be determined by matching the solutions (21) with Schwarzschild exterior solution for a ball of mass 𝑀 and linear dimension 2𝑟_𝑏:𝑒𝑝𝑟_𝑏=0,_{−𝜆(𝑟𝑏)}𝑒=1−2𝑢,_{𝜐(𝑟𝑏)}=1−2𝑢,(25) where𝑢=𝐺𝑀𝑐²𝑟_𝑏.(26) Consequently,𝐶₁=−[(𝑎−𝑏−1)(1−2𝑢)−1]2𝑏𝑟^{𝑏𝑎+𝑏−2}𝐶,(27)₂=[(𝑎+𝑏−1)(1−2𝑢)−1]2𝑏𝑟^{𝑏𝑎−𝑏−2}𝐴=𝑟_𝑏𝑙−7,(28)[1−2𝑢](𝑎+𝑏)𝐶¹𝑟^𝑏2𝑏+(𝑎−𝑏)𝐶₂𝐶₁𝑟^𝑏2𝑏+𝐶₂+𝑙−47.(29)

For 𝑒^𝜐 to be definitely positive in the region 0≤𝑟≤𝑟_𝑏, we must have 𝐶₁,𝐶₂≥0. Thus in view of (27) and (28), we have𝑎+𝑏−2𝑎+𝑏−1≤2𝑢≤𝑎𝑎−−𝑏𝑏−−12.(30)

5. Particular Member of Class I for 𝑙=4.75

In this section, we shall present a detail study of the particular solution corresponding to 𝑙=4.75.

The solution is ^{𝑒𝜐=𝐶1𝑟0.35962+𝐶2𝑟1.39,𝑒−𝜆=𝐴𝑟2.25𝐶+1.781+𝐶2𝑟1.032.359𝐶1+3.39𝐶2𝑟1.03,8𝜋𝐺}_{𝑐41𝑝=𝑟2𝐴𝑟}^{2.25+1.781.359𝐶1+2.39𝐶2𝑟1.03}_{2.359𝐶1+3.39𝐶2𝑟1.03,−18𝜋𝐺𝑐21𝜌=𝑟2𝐴𝑟2.25+1.781.06𝐶1𝐶2𝑟1.03+−3.25𝐴𝑟2.25×𝐶−1.781+𝐶2𝑟1.032.359𝐶1+3.39𝐶2𝑟1.032.359𝐶1+3.39𝐶2𝑟1.032.+1}(31) In view of (30) 𝑒^𝜐 to be definitely positive in the region 0≤𝑟≤𝑟_𝑏, then0.13225≤𝑢≤0.29079.(32) Corresponding to 𝑢=0.25 and in view of (27), (28), and (29), the constants are𝐶₁𝑟^𝑏0.35962=𝐴0𝑟.18_𝑏2.2591,𝐶₂𝑟^𝑏1.39=0.3106,=−0.2808.(33)

In Table 1 the march of pressure, density, pressure-density ratio, and square of adiabatic sound speed 𝑑𝑝/𝑑𝜌 is given for 𝑢=0.25. We observe that pressure and density decrease monotonically with the increase of radial coordinate, pressure-density ratio, and square of adiabatic sound speed, which is positive and less than 1 throughout within the ball.

Table 1 

The march of pressure, density, pressure-density ratio, and square of adiabatic sound speed within the ball corresponding to 𝑙=4.75 with 𝑢=0.25.

We now present here a model of Neutron star based on the particular solution discussed above. The Neutron star is supposed to have a surface density: 𝜌_𝑏=2×10¹⁴gm-cm⁻³ suggested by Brecher and Caporaso [13]. The resulting causal model has the mass 𝑀=2.40𝑀_Θ and the linear dimension 2𝑟_𝑏≈28.43km. The surface red shift 𝑍_𝑏=[(1−2𝑢)^−0.5−1]≈0.4142.

6. Some More Parametric Classes of Solutions

By assigning the different nonnegative integral values to the exponents of (20), we obtain the following different parametric classes of solutions.

6.1. Class II

If we assume 𝛼−1=0 and −1+2/𝑚=0. In view of (17), (18), and (20), the resulting class of solutions is (see Neeraj Pant [6])𝑒𝑒^𝜐_−𝜆=𝐶₁𝑟^{𝑎+𝑏−2}+𝐶₂𝑟^{𝑎−𝑏−2},=𝐴𝑟^−2𝑏(𝑎+𝑏)𝐶₁𝑟^2𝑏+(𝑎−𝑏)𝐶₂^𝑛+2−𝑟^−2𝑏2𝑏(𝑛+2)(𝑎+𝑏)𝐶₁𝐶¹𝑟^2𝑏+𝐶₂.(34)

6.2. Class III

If we assume 𝛼−1=0 and 𝑛+1=0. In view of (17), (18), and (20), the resulting class of solutions is (see Pant [8])𝑒𝑒^𝜐_−𝜆=𝐶₁𝑟^{𝑎+𝑏−𝑚}+𝐶₂𝑟^{𝑎−𝑏−𝑚}^2/𝑚,=𝐴𝑟^−2𝑏−𝑚²𝑟^−2𝑏/2𝑏𝐶₁C₁𝑟^2𝑏+C₂^2/𝑚(𝑎+𝑏)𝐶₁𝑟^2𝑏+(𝑎−𝑏)𝐶₂C₁𝑟^2𝑏+C₂^{2(−1+1/𝑚)}.(35)

6.3. Class IV

If we assume 𝑛+1=0 and −1+2/𝑚=1. In view of (17), (18), and (20), the resulting class of solutions is (see Pant [8])𝑒𝑒^𝜐_−𝜆=𝐶₁𝑟^{𝑎+𝑏−1}+𝐶₂𝑟^{𝑎−𝑏−1}²,=𝑟^−2𝑏𝛼𝐴−𝑟^2𝑏𝛼𝛼𝐶₁𝑟^2𝑏+(𝛼+1)𝐶₂/𝑏𝛼(𝛼+1)(𝑎+𝑏)𝐶₁𝑟^2𝑏+(𝑎−𝑏)𝐶₂,(36) where𝛼=𝑙−7+𝑎−𝑏2𝑏.(37)

6.4. Class V

If we assume 𝛼−1=1 and −1+2/𝑚=0. In view of (17), (18), and (20), the resulting class of solutions is (New Class of Solutions)𝑒𝑒^𝜐_−𝜆=𝐶₁𝑟^{𝑎+𝑏−2}+𝐶₂𝑟^{𝑎−𝑏−2},=𝐴𝑟^−4𝑏−4𝐵𝑟^−4𝑏𝔄𝔅(𝑎+𝑏)𝐶₁𝑟^2𝑏+(𝑎−𝑏)𝐶₂^𝑛+2𝐶₁𝑟^2𝑏+𝐶₂⁻¹,(38) where 𝔄 denotes [(𝑎+𝑏)𝐶₁𝑟^2𝑏+(𝑎−𝑏)𝐶₂]^𝑛+2 and 𝔅 denotes [(𝑎+𝑏)(𝑛+2)𝐶₁𝑟^2𝑏−(𝑎−𝑏)𝐶₂], 𝐵=1𝑏(𝑎+𝑏)²(𝑛+2)(𝑛+3)𝐶₁₂.(39)

6.5. Class VI

If we assume 𝛼−1=1 and 𝑛+1=0. In view of (17), (18), and (20), the resulting class of solutions is (New Class of Solutions)𝑒𝑒^𝜐_−𝜆=𝐶₁𝑟^{𝑎+𝑏−𝑚}+𝐶₂𝑟^{𝑎−𝑏−𝑚}^2/𝑚,=𝔇2𝐶¹𝑟^2𝑏−𝑚𝐶₂(𝑎+𝑏)𝐶₁𝑟^2𝑏+(𝑎−𝑏)𝐶₂C₁𝑟^2𝑏+C₂^{2(−1+1/𝑚)},(40) where 𝔇 denotes 𝐴𝑟^−4𝑏−2𝑚²𝑟^−4𝑏/4𝑏𝐶₁₂(C₁𝑟^2𝑏+C₂)^2/𝑚.

6.6. Class VII

If we assume 𝛼−1=0 and −1+2/𝑚=1. In view of (17), (18), and (20), the resulting class of solutions is (see Tewari and Pant [11])𝑒𝑒^𝜐_−𝜆=𝐶₁𝑟^{𝑎+𝑏−1}+𝐶₂𝑟^{𝑎−𝑏−1}²,=𝐴𝑟^−2𝑏(𝑎+𝑏)𝐶₁𝑟^2𝑏+(𝑎−𝑏)𝐶₂^𝑛+2−𝐵𝐶_1/+𝐶_2/𝑟^−2𝑏.(41)

7. Conclusion

By assigning different nonnegative integral values to the any two pairs of exponents of integrand of (17), that is, 𝛼−1,𝑛+1,−1+2/𝑚, we arrive at different parametric classes of solutions. All these classes have of the form 𝑒^𝜐=(𝐶₁𝑟^𝛽+𝐶₂𝑟^𝛽)^𝛾.For meaningful solutions 𝛾, 𝛽 to be nonnegative, thus restriction on parameter ranges.

Acknowledgments

The author expresses his gratitude to Professer O. P. Shukla, Principal NDA, for his encouragement. He is also grateful to the anonymous referees for pointing out the errors in the original paper and making constructive and relevant suggestions.