Isotropic cases of static charged fluid spheres in general relativity
In this paper we study the isotropic cases of static charged fluid spheres in general relativity. For this purpose we consider two different specialization and under these we solve the Einstein-Maxwell field equations in isotropic coordinates. The analytical solutions thus we obtained are matched to the exterior Reissner-Nordström solutions which concern with the values for the metric coefficients and . We derive the pressure, density, pressure-to-density ratio at the centre of the charged fluid sphere and boundary of the star. Our conclusion is that static charged fluid spheres provide a good connection to compact stars.
The topic of static charged fluid spheres in general relativity is a challenging issue and has given rise to interesting studies, even if their number is not so high. Considering the research of the interior of stars in connection to their late stage evolution when the general relativistic effects play an important role, we notice the interesting work of Tolman  that yields a class of solutions for the static, spherically symmetric equilibrium fluid distribution. This important result was followed by some generalizations made by Wyman , Leibovitz  and Whitman . After them, Bayin  used the idea of the method of quadratures and gave new astrophysical solutions for the static fluid spheres.
In recent years, Ray and Das [6, 7, 8] and Ray et. al.  have developed some interesting solutions for the static charged fluid spheres in general relativity following the above line of thinking. Ray and Das  performed the charged generalization of Bayin’s work  related to Tolman’s type astrophysically interesting aspects of stellar structure . In this light, they actually considered that even in the case of stellar astrophysics there are physical implications of the Einstein-Maxwell field equations. In other papers Ray and Das [7, 8] performed a study of some previous solutions considering the phenomenological connection of the gravitational field to the electromagnetic field, and demonstrated the purely electromagnetic origin of the charged relativistic stars given by Tolman  and Bayin . The existence of this type of astrophysical solutions is thought to be a probable extension of Lorentz’s conjecture  that electron-like extended charged particle possesses only ‘electromagnetic mass’ and no ‘material mass’. In their latest work Ray et al.  have considered Tolman-Bayin type static charged fluid spheres in general relativity and they have found out many interesting results, specially the cases which give support to the charged spherical models in connection to normal stars. It is argued by them  that due to the inclusion of charge and by a suitable choice of charge part, pressure and density function could be a decreasing function of radius from centre to surface in contrary to Bayin’s case .
In this connection we also notice other two works, the first one elaborated by Varela , which presents a neutral perfect fluid core bounded by a charged thin shell, and which demonstrates that it is possible to construct extended Reissner-Nordström sources with everywhere positive mass density, classical electron radius, electromagnetic mass, and everywhere non-negative gravitational mass. The other work was done by Ivanov  that studied the interior perfect fluid solutions for the Reissner-Nordström metric using a new classification scheme. In addition, he found general formulae in some particular cases, presented explicit new global solutions and made a revision of some known solutions. In connection to the singularity problem it is argued by Ivanov  that the presence of the charge function serves as a safety valve, which absorbs much of the fine-tuning necessary in the uncharged case. However, it is also believed by some authors [13, 14] that in the presence of charge the gravitational collapse of a spherically symmetric distribution of matter to a point singularity may be avoided through counterbalancing of the gravitational attraction by the repulsive Coulombian force in addition to the thermal pressure gradient due to fluid.
In continuation of the above theme, especially that of Ray et al. , in the present article we have tried to solve the Einstein-Maxwell field equations in isotropic coordinate system and derived expressions for pressure and density. We have found out conditions for the boundary of the charged sphere. The exterior Reissner-Nordström solution is compared and constants of integrations are expressed in terms of mass and radius. The mass-radius and mass-charge relations have been found out for various cases of the charged matter distribution along with the pressure-to-density ratio at the centre of the charged sphere.
Our paper is organized as follows: in Section 2 we introduce the
Einstein-Maxwell field equations for the static charged fluid
spheres in general relativity, we determine their analytical
solutions making some assumption for the metric coefficients
and and we consider two particular cases. In
Section 3 we find out boundary conditions, which consist in a
vanishing value for the pressure and a specific value for the
boundary of the star. The expression for the radius of the
star is established in Section 4 that is dedicated to a
detailed analysis of the obtained solutions. The role of some
parameters and integration constants in the evolution of pressure
, density and radius of the star is discussed, with
emphasis on some particular values. In Conclusions we enlighten
the physical significance of the results.
2 The Einstein-Maxwell field equations and their analytical solutions
The static spherically symmetric matter distribution corresponding to the isotropic line element is given by
where and are functions of the radial coordinate only.
The Einstein-Maxwell field equations for the above line element can be written as
where the total charge with a sphere of radius in terms of the -currents is
being the intensity of the electric field. Here and are the pressure and matter-energy density, respectively. We have used prime to denote derivative with respect to radial coordinate only. The field equations without the charge have been studied by Buchdahl  and Bayin  who found physically meaningful solutions.
where and .
Hence, to solve the above equation (6) for we make use of the ansatz so that the above equation becomes
with where behaves as the polynomial index. A direct
solution for the above second order differential equation is
difficult to find out. However, we can set different conditions
for the parameters to get a solvable equation. Let us therefore
consider the following two cases.
2.1 The case for and
For the specification , when , we get the following solutions
2.2 The case for and
For this choice of and the solutions set becomes
We notice from the above two subcases that and
represent integration constants and their expressions will be
determined in Section in terms of and .
3 Boundary conditions
The exterior field of a spherically symmetric static charged fluid distribution described by the metric (1) in isotropic coordinates is the unique Reissner-Nordström solution
which by the radial coordinate transformation 
takes the form
Therefore, matching at we get
where is the boundary of the star. At the boundary pressure is
zero. Using this we will get an expression for in the next
4 The study of the solutions
4.1 The case for and
In Fig. 1, Fig. 2 and Fig. 3 we plot the , and against the parameter for some fixed values of the integration constants and and parameters , , , and .
Fig. 4 above indicates that is positive. That means though that the Weak Energy Condition (WEC) is violated in our case but obeys the Null Energy Condition (NEC) as well as the Strong Energy Condition (SEC) and the Dominant Energy Condition (DEC). This is a very specific charged fluid solution. In reference to the figure for p (Fig. 1), since gives the radius of the charged fluid (rather we would say, ), one may conclude that the radius of the charged fluid falls within to Km. Thus our charged fluid is highly compact and we may consider our charged fluid as typically a highly compact strange star (we call it ‘strange’ in the sense that EOS components follow peculiar properties here i.e. they obey all energy conditions except WEC). In this context we would like to mention that in the original Bayin solutions, case V and case VI, is also negative.
Therefore, following Bayin  we are now in the position to find out pressure-to-density ratio at the centre of the charged sphere in the form
We notice that the pressure-to-density ratio at the centre of the charged sphere is entirely independent of the and parameters and integration constant . In the case with we have and this is not the case with simultaneous positive pressure and density at the origin. For and we obtain , which is a physically meaningful result for positive values for pressure and density at the origin. As a special case, if we choose , then we can easily recover the equation of state related to the Cases I and II for of Ray et al.  which refers to the ‘false vacuum’ or ‘degenerate vacuum’ or ‘-vacuum’ equation of state [17, 18, 19, 20]. Due to the repulsive nature of the pressure this provides a mechanism to avert the problem of singularity at the centre. On the other hand, if we choose and then we can recover the result of Ray et al.  related to the Case I for which refers to radiation. In Table 1 we have shown different possibilities for different values of the parameters and .
Substituting , we can get an expression for the boundary
of the charged sphere as follows:
Though it is a complicated power law equation for , and hence very difficult to solve yet from this we get the same expression which was obtained by Bayin (1978) in his non-charge case (see equation (4.23) there in).
Again, comparing Reissner-Nordström metric in isotropic coordinates (16) [and hence (17) and (18)] we get
Therefore, for we get
Therefore, considering arbitrary values for , and we get expressions for , and in terms of and .
4.2 The case for and
The pressure-to-density ratio at the centre of the charged sphere in this case is as follows :
The same ‘false vacuum’ or ‘degenerate vacuum’ or ‘-vacuum’ equation of state [17, 18, 19, 20] is achieved again without imposing the condition in the present case. Therefore, also due to the repulsive nature of the pressure this provides a mechanism to avert the problem of singularity at the centre.
The radius of the star is given by the expression obtained from the condition . However, here and have to be taken at , so that the final expression for the radius can be written as
where we have used the symbol .
We can find out the values of , and in terms of and from the equation (37) and the following boundary conditions
The study of static charged fluid spheres in general relativity is far from being complete, but some recent studies have motivated us to extend one previous work  for improving the understanding of this topic. We found out the solutions of field equations in isotropic cases of charged fluid spheres in general relativity and performed a detailed study of the analytical solutions. In order to obtain the analytical solutions of the Einstein-Maxwell field equations, we made use of some assumptions and introduced two specialization, the first with the specification and and the second with the values and . We calculated the corresponding expressions for the function and for the metric coefficients and , which depend on the parameters , , , and integration constants and . Matching the interior solution to the exterior Reissner-Nordström metric in isotropic coordinates at we obtained the formulae for and .
We performed a deeper investigation of the solutions and for the case and we established the expression for the pressure-to-density ratio at the origin given by and the imposed conditions for and for obtaining a physically meaningful model for the star. For a zero value of we obtained the case of the false vacuum or degenerate vacuum or -vacuum [17, 18, 19, 20]. In addition, we found out the explicit expression for the radius and discussed the physically reasonable cases. The radius for the charged sphere has a completed expression which, however, reduces to the same expression as in the non-charge case of Bayin . Finally, comparing Reissner-Nordström metric in isotropic coordinates we established the expressions for the integration constants and and parameter in terms of mass and the radius of the star . On the other hand for the specification and we computed the pressure and the density that present a dependence on a new function and its first and second derivatives with respect to the coordinate. In this case the expression for the pressure-to-density ratio at the origin can directly be given by . We also gave the expressions that allow to evaluate the boundary of the star and the values of , and in terms of and , but due to their complicated form we plan to perform a detailed study in a future work.
The most straight forward observation in the present work is that
from our results we are able to recover the neutral cases of Bayin
. We also observe that the models presented here
have negative energy density and positive pressure and support the
conclusion that static charged fluid spheres are connected to
compact strange stars. We conclude that in the last years there is
progress in the study of static charged fluid spheres which are
connected to compact stars, but deeper and wider investigations
for searching for new solutions need to be performed. One
possibility will be to investigate our scenario based on different
specialization and obtain new analytical solutions that can lead
to other particular cases and more constrained connections with
the involved parameters.
FR and SR are thankful to the authority of Inter-University Centre for Astronomy and Astrophysics, Pune, India for providing research facilities.
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