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In the present study, we have obtained a new analytical solution of combined Einstein-Maxwell field equations describing the interior field of a ball having static spherically symmetric isotropic charged fluid within it. The charge and electric field intensity are zero at the center and monotonically increasing towards the boundary of the fluid ball. Besides these, adiabatic index is also increasing towards the boundary and becomes infinite on it. All other physical quantities such as pressure, density, adiabatic speed of sound, charge density, adiabatic index are monotonically decreasing towards the surface. Causality condition is obeyed at the center of ball. In the limiting case of vanishingly small charge, the solution degenerates into Schwarzchild uniform density solution for electrically neutral fluid. The solution joins smoothly to the Reissner-Nordstrom solution over the boundary. We have constructed a neutron star model by assuming the surface density . The mass of the neutron star comes with radius 14.574 km.

An analysis of the Reissner-Nordstrom metric shows that a spherically symmetric distri- bution of charged dust may avoid the catastrophic gravitational collapse, a seemingly unavoidable feature of Schwarzschild’s [

The search for the exact solutions is of continuous interest to researcher. Buchdahl [

In this paper, we present a new solution of Einstein-Maxwell field equations in sphe- rically symmetric coordinates which are well behaved solutions charge analogous solution of Sah and Chandra [

The Einstein-Maxwell field equations in general relativity are given by

where

The energy momentum tensor for a charged fluid sphere is defined as

Here

where

where the electromagnetic tensor

and is given by

The total non gravitational energy in a proper volume V is given by

We consider a static spherically symmetric charged perfect fluid distribution. The interior space-time metric for spherically symmetric fluid distribution in canonical coordi- nate is given by

where

The electrostatic field is described by the only non-singular components

In view of (8), (9) and (10) we obtain

where Q stands for the total charge contained within the sphere of radius r and is given as

In view of the metric (14) and energy momentum tensor (3), the field Equation (1) gives

In view of Equation (17) and Equation (18), pressure isotropy gives

The charge conservation Equation (10) is identically satisfied whereas (2) for energy- momentum gives rise to the following surviving equation

which is clearly contained in the field Equations (17) to (20).

We have three equations to determine five unknown functions

where M and E are constants. We observe that whereas in Schwarzschilds field that total energy

approximates to Schwarzschilds field. The junction of interior and exterior field over the boundary

Thus the constant

Also we have

where

and

The continuity of

Thus, in general, the constant M can not be identified with Euclidean mass of the sphere as against the case of uncharged sphere in which case

Thus three physical quantities contribute to

The gravitational redshift of massive spherically symmetric ball is

which gives central

and

In view of (11) the total non gravitational energy of a charged sphere is given

Also the total energy of the ball measured by an electrically neutral test particle close to the boundary is given by

Clearly the difference

For a particle at large distances from the object total energy

expression for the energy of vacuum electrostatic field surrounding a sphere of charge

For well behaved nature of the solution in isotropic coordinates, the following conditions should be satisfied:

1) The solution should be free from geometrical and physical singularities. Metric potentials A and B must be non-zero positive finite for free from geometrical singularities while central pressure, central density, should be positive and finite or

2) The solution should have maximum positive values of pressure and density at the center and monotonically decreasing towards the surface of fluid object i.e.

i)

ii)

3) At boundary pressure,

4) The pressure,

5) Solution should have positive value of pressure-density ratio which must be less

than 1 (weak energy condition) and less than

within the fluid object and monotonically decreasing as well (Pant and Negi [

6) The casualty condition must be satisfied for this velocity of sound should be less

than that of light throughout the model i.e.

be monotonically decreasing towards the surface and increasing with the increase of

density i.e.

the equation of state at ultra-high distribution has the property that the sound speed is decreasing outwards.

7) For realistic matter, the adiabatic index

ball.

8) The red shift at the center

Under these conditions, we have to assume the one of the gravitational potential component in such a way that the field Equation (1) can be integrated and solution should be well behaved.

We present the following general analytic solution of the field Equations (17) to (20).

The isotropic pressures, matter-energy density, charge, charge density and red shift of charged fluid ball are given by

Here a, b, and d are arbitrary constants.

In order to construct a new relativistic model, we assume

Here a, b, and d are arbitrary constants.

The variation in pressure, density, charge, charge density and red shift with radial distance are given as

For real values of metric potentials A and B,

The central value of

It is clear from Equation (53) to Equation (57) that for positive central values of physical quantities

with the distance from the center of fluid ball are identically zero at the center.

At the center of fluid ball the second order derivatives of pressure and density with respect to radial distance from the center of fluid ball are

The pressure is maximum at the center if

The density is maximum at the center for all constants as

The central equation of state

The central value of

The causality condition at the center

It is found that

boundary of the fluid ball.

The solution so obtained are to be matched over the pressure free boundary of fluid sphere smoothly with the Reissner-Nordstrom metric:

which requires the continuity of

where

In view of Equations (64) to (66) the values of

0.0823c and −0.0753 respectively and the value of

In order to construct a super dense star model, we prescribe the surface density of the star as

1 | 0.0 | 0.76085 | 1.00000 | 1.02000 | 0.02768 | 0.02715 |

2 | 0.1 | 0.76155 | 1.00170 | 1.01762 | 0.02737 | 0.02690 |

3 | 0.2 | 0.76365 | 1.00683 | 1.01048 | 0.02644 | 0.02617 |

4 | 0.3 | 0.76719 | 1.01547 | 0.99862 | 0.02490 | 0.02493 |

5 | 0.4 | 0.77219 | 1.02776 | 0.98210 | 0.02277 | 0.02319 |

6 | 0.5 | 0.77871 | 1.04389 | 0.96099 | 0.02009 | 0.02091 |

7 | 0.6 | 0.78682 | 1.06413 | 0.93541 | 0.01690 | 0.01806 |

8 | 0.7 | 0.79661 | 1.08881 | 0.90546 | 0.01323 | 0.01462 |

9 | 0.8 | 0.80818 | 1.11837 | 0.87129 | 0.00915 | 0.01050 |

10 | 0.9 | 0.82169 | 1.15335 | 0.83303 | 0.00471 | 0.00565 |

11 | 1.0 | 0.83729 | 1.19442 | 0.79085 | 0.00000 | 0.00000 |

1 | 0.0 | 0.13119 | 4.83330 | 0.88334 | 0.00000 | 0.31432 |

2 | 0.1 | 0.13099 | 4.86945 | 0.88107 | 0.00021 | 0.31311 |

3 | 0.2 | 0.13037 | 4.98243 | 0.87412 | 0.00085 | 0.30948 |

4 | 0.3 | 0.12935 | 5.18736 | 0.86295 | 0.00188 | 0.30345 |

5 | 0.4 | 0.12791 | 5.51540 | 0.84794 | 0.00331 | 0.29501 |

6 | 0.5 | 0.12605 | 6.02752 | 0.82972 | 0.00512 | 0.28416 |

7 | 0.6 | 0.12376 | 6.84985 | 0.80905 | 0.00728 | 0.27093 |

8 | 0.7 | 0.12104 | 8.28097 | 0.78683 | 0.00979 | 0.25532 |

9 | 0.8 | 0.11788 | 11.22241 | 0.76402 | 0.01265 | 0.23733 |

10 | 0.9 | 0.11427 | 20.20590 | 0.74168 | 0.01582 | 0.21700 |

11 | 1.0 | 0.11020 | 0.72088 | 0.01943 | 0.19433 |

We have given a new solution for spherically symmetric isotropic charged fluid ball. It has been observed that the physical parameters pressure, density, adiabatic speed of sound and redshift are positive at the centre and within the limit of realistic state equation and monotonically decreasing and the causality condition is obeyed through- out the fluid ball. The charge and electric field intensity are zero at the center and monotonincally increasing towards the intervening surface. Thus, the solution is well behaved for all values of Schwarzschild parameter

1 | 0.080 | 0.017 | 0.150 | 0.7164 | 0.765 | 14.330 | 1.451 |

2 | 0.086 | 0.019 | 0.158 | 0.6978 | 0.810 | 14.750 | 1.573 |

3 | 0.089 | 0.021 | 0.167 | 0.6887 | 0.831 | 14.940 | 1.684 |

4 | 0.092 | 0.022 | 0.171 | 0.6797 | 0.851 | 15.118 | 1.745 |

5 | 0.096 | 0.024 | 0.178 | 0.6678 | 0.873 | 15.312 | 1.840 |

6 | 0.101 | 0.026 | 0.186 | 0.6531 | 0.904 | 15.582 | 1.956 |

7 | 0.106 | 0.029 | 0.195 | 0.6387 | 0.936 | 15.855 | 2.087 |

8 | 0.110 | 0.031 | 0.202 | 0.6274 | 0.961 | 16.065 | 2.190 |

9 | 0.115 | 0.034 | 0.209 | 0.6134 | 0.988 | 16.289 | 2.290 |

10 | 0.120 | 0.036 | 0.218 | 0.5996 | 1.017 | 16.527 | 2.432 |

11 | 0.126 | 0.040 | 0.228 | 0.5835 | 1.048 | 16.777 | 2.582 |

12 | 0.130 | 0.042 | 0.235 | 0.5728 | 1.070 | 16.952 | 2.689 |

13 | 0.134 | 0.045 | 0.242 | 0.5624 | 1.091 | 17.118 | 2.796 |

14 | 0.140 | 0.049 | 0.251 | 0.5470 | 1.119 | 17.336 | 2.937 |

15 | 0.145 | 0.052 | 0.259 | 0.5344 | 1.141 | 17.505 | 3.060 |

Neutron star and many more. We have discussed a model of massive neutron star having mass

Sah, A. and Chandra, P. (2016) Class of Charged Fluid Balls in General Relativity. International Journal of Astronomy and Astrophysics, 6, 494-511. http://dx.doi.org/10.4236/ijaa.2016.64038