A family of solutions to the Einstein-Maxwell system of equations describing relativistic charged fluid spheres
In this paper, we present a formalism to generate a family of interior solutions to the Einstein-Maxwell system of equations for a spherically symmetric relativistic charged fluid sphere matched to the exterior Reissner-Nordström spacetime. By reducing the Einstein-Maxwell system to a recurrence relation with variable rational coefficients, we show that it is possible to obtain closed-form solutions for a specific range of the model parameters. A large class of solutions obtained previously are shown to be contained in our general class of solutions. We also analyze the physical viability of our new class of solutions.
1 Department of Mathematical Sciences, Faculty of Applied Sciences, South Eastern University, Sri Lanka.
\affilTwo2 Department of Physics, P. D. Women’s College, Jalpaiguri 735101, India.
Exact solutions to Einstein-Maxwell (EM) system of field equations play an important role in the studies of self-gravitating spherically symmmetric charged fluid distributions. Ever since the discovery of the Reissner-Nordström solution, many investigators have contributed to the study of EM system which includes the pioneering works of Papapetrou , Majumdar , Bonner [3, 4], Stettner , Bekenstein  and Cooperstock and Cruz . A detailed review of exact solutions to Einstein-Maxwell systems and their physical acceptability can be found in the compilation work of Ivanov . A large class of interior solutions, corresponding to the exterior Reissner-Nordström spacetime, have been developed to model a wide variety of stellar distributions such as neutron stars, strange stars and stellar objects composed of quark-diquark mixtures [9, 10, 11, 12, 13, 14, 15, 16, 17]. Stellar models have also been developed for charged core-envelope type configurations [18, 19, 20]. Mak and Harko  and Komathiraj and Maharaj  have obtained solutions for charged strange quark stars admitting a linear equation of state (EOS). Thirukkanesh and Maharaj  have analyzed the role of anisotropy on the physical behaviour of a given charged distribution admitting a linear EOS. Varela et al  have analyzed features of a charged anisotropic fluid distribution admitting linear as well as non-linear EOS. Takisa and Maharaj  have obtained a new class of solutions for a charged quark matter distribution. Feroze and Siddiqui  and Maharaj and Takisa  have independently developed charged stellar models by assuming a quadratic EOS. Thirukkanesh and Ragel [28, 29] have obtained new solutions for charged fluid spheres by specifying the polytropic index leading to masses and energy densities which have been shown to be consistent with observational data. Maharaj et al  have presented a new family of exact solutions to the Einstein-Maxwell system for an anisotropic charged matter on the Finch and Skea  background spacetime. A class of charged anisotropic stellar solutions has been developed and studied by Murad and Fatema . Hansraj et al  have analyzed all static-charged dust sphere models in general relativity. Recently, Sunzu and Danford  have generated two new class of exact solutions to the Einstein-Maxwell system of field equations describing an anisotropic and charged stellar body which accommodates a quark matter like linear EOS.
The main objective of the present work is to contribute to this rich family of solutions by generating new solutions which can be used as viable models of realistic astrophysical objects. While generating the solutions, one needs to ensure that the gravitational, electromagnetic and matter variables remain finite, continuous, well behaved and the speed of sound remains less than the speed of light within the distribution. For a charged fluid sphere, the interior solution must be matched to the exterior Riessner-Nordström metric across the boundary. We present here a different family of solutions to the coupled Einstein-Maxwell system where all the above requirements are fulfilled. This has been done by choosing a rational form for one of the gravitational potentials and also the fall-off behaviour of the charged fluid distribution. This particular approach is similar to the method adopted earlier by Maharaj and Leach  which was, in fact, a generalization of the superdense stellar model developed by Tikekar . In our approach, the solutions are generated by reducing the condition of pressure isotropy to a recurrence relation with real and rational coefficients so that the system can be solved by mathematical induction. We have performed a systematic analysis of the new family of solutions to examine their physical viability.
The paper has been organized as follows: In Section , we have presented the EM field equations for a static spherically symmetric charged fluid distribution. The nonlinear system was then transformed into a more tractable set of equations. By assuming a particular form for one of the metric potentials and also by specifying the electric field intensity, we have obtained the condition of pressure isotropy in terms of the undetermined gravitational potential in Section . We have assumed a series solution for the resultant equation which yielded a recurrence relation. We have managed to solve the system from the first principles in Section . In Section , we have presented polynomials and product of polynomials with algebraic functions as the first solution. The general solution containing the integral form was eventually integrated to yield elementary functions by placing specific restrictions on the model parameters. We have demonstrated that it is possible to regain many solutions found earlier by adopting this technique. Finally, we have provided two different class of exact solutions to the EM system in simple closed forms. In Section , we have discussed features of the class of solutions and showed that the solutions might be used to model realistic compact stellar systems. The results have been summarized in Section .
2 Einstein-Maxwell system
For a static spherically symmetric relativistic charged fluid distribution, we assume the line element in coordinates as
where and are arbitrary functions of the radial coordinate . The Einstein-Maxwell field equations for the line element (1) are then obtained (in system of units having 8 = = 1) as
where a prime ( ) denotes differentiation with respect to . The energy density and the pressure are measured relative to the comoving fluid 4-velocity . The electric field intensity and the proper charge density appear into the system through the energy-momentum tensor corresponding to the electro-magnetic field and the Maxwell equations.
A different but equivalent form of the field equations can be generated if we introduce a new independent variable and introduce new functions and
where dots (.) denote differentiation with respect to the variable . The system (4) determines the gravitational behaviour of a charged perfect fluid. Consequently, we have a nonlinear system of four independent equations in six unknowns variables namely, , , , , and . The advantage of this system lies in the fact that a solution, upon suitable substitutions of and , can be obtained by integrating the second order differential equation (4c) which is linear in .
3 Integration procedure
We solve the Einstein-Maxwell system (4) by making explicit choices for the metric function and the electric field intensity . For the metric function we write
where and are real constants. Note that the choice (5) ensures that the metric function is regular and continuous in the interior because of the freedom provided by parameters and . It is important to note that the particular choice of is physically reasonable and contains some special cases of known relativistic star models. The case corresponds to the Maharaj and Komathiraj  charged stellar model which is a generalization of the stellar models developed previously by Finch and Skea  and Hansraj and Maharaj . A similar form of has also been utilized in Ref. [26, 27, 40] for the construction of a charged anisotropic stellar model admitting a polytropic EOS.
It is convenient at this point to introduce the following transformation
This transformation enables us to rewrite the second order differential equation (3) in a simpler form
in terms of the new dependent and independent variables and , respectively. To integrate (8), it is necessary to specify the electric field intensity . Even though a variety of choices for is possible, only a few of them are physically reasonable and can generate closed form solutions. We reduce (8) to an integrable form by letting
where and are constants. The form in (9) is physically acceptable as remains regular and continuous throughout the sphere. Note that at . Some special cases of (9) have earlier been studied by Takisa and Maharaj  and John and Maharaj  and can also be reduced to the uncharged stellar model developed by Maharaj and Mkhwanazi . Substituting (9) in equation (8), we obtain
which is the limiting case and corresponds to an uncharged sphere.
4 General series solution
A closed form solution of equation (10) is difficult to obtain. However, one can transform it to a differential equation which can be integrated by the method of Frobenius. This can be done in the following way.
We introduce a new function such that
where is a constant. A similar kind of transformation was utilized earlier by Komathiraj and Maharaj  for generating charged stellar models. With the help of (12), the differential equation (10) can be written as
A substantial simplification of the equation can be achieved if we set
Equation (13) then reduces to
where we have set
Since the point is a regular singular point of (15), there exists two linearly independent solutions of the form of power series with centre . Therefore, we can write the solution of the differential equation (15) by the method of Frobenius as:
where are the coefficients of the series and is a constant.
and the recurrence formula
with and . Since , , we must have or . The coefficients can all be written in terms of the leading coefficient and we can generate the expression as
It is also possible to establish the result (19) rigorously by using the principle of mathematical induction.
where and For the parameter value , we obtain the second solution
where and We, therefore, have a general solution to (15) as the functions and are linearly independent. In terms of the original variable , the functions are obtained as
As the choice of the metric function (5) together with the electric field intensity (9) have not been considered earlier, to the best of our knowledge, the class of solutions (23) have not been reported previously. One interesting feature of the new family of solutions is that by setting and , it is possible to switch off the effect of charge onto the system. Secondly, the solution (22) has been expressed in terms of a series of real arguments and not complex arguments which one might encounter when mathematical software packages are used.
5 Terminating series
It is interesting to observe that the series in (20) and (21) terminates for specific values of the parameters and . It is, therefore, possible to generate solutions in terms of elementary functions by imposing specific restrictions on and . The solutions may be found in terms of polynomials and algebraic functions. We use recurrence relation (18), rather than the series (20) and (21), to find the elementary solutions.
5.1 Elementary solutions
If we fix in (18), and set , for integer values of , we obtain
where is a fixed integer. Obviously, . Consequently, the remaining coefficients vanish. Equation (24) may be solved to yield
On substituting in (18) and by setting , we obtain
where is a fixed integer. Obviously, and the subsequent coefficients vanish. Equation (27) yields
where and . The polynomial (26) and the product of polynomial and algebraic function (29) generate a particular solution of the differential equation (15) for appropriate values of the parameters and .
5.2 General solutions
It is possible to obtain solutions to (15) by restricting the values of and so that only elementary functions survive. The elementary functions are expressible as polynomials and product of polynomials with algebraic functions. Using (26), we express the first category of general solutions to the differential equation (15) in the form
where . Equation (5.2) in terms of , takes the form
where and .
We, thus have generated two class of solutions (5.2) and (5.2) to the differential equation (3) for the assumed electric field (9) making use of the infinite series solution (22). It should be stressed that the class of solutions can be used to study stellar properties in the presence as well as absence of charge. By setting and in (5.2) and (5.2), one obtains solutions for an uncharged sphere.
By setting and in (34), we obtain
where we have assumed and . Thus, we have regained the Durgapal and Bannerji solution .
If we set and in (35), we obtain
where we have assumed and . This class of solutions was found earlier by Maharaj and Mikhwanazi .
5.3 New family of solutions
We now aim to generate new closed form solutions for which can subsequently be used to model realistic stars. To achieve our objective we set and . Then, making use of (34), it is possible to obtain two categories of solutions for (i) and (ii)
In this case (34) becomes
where we have set
Subsequently, the general solution to the Einstein-Maxwell system (23) can be expressed as
In this case (34) becomes
where we have set
The simple form of our class of solutions facilitates the analysis of matter and gravitational variables of realistic stellar objects as can be seen in the following sections.
6 Physical analysis
For a physically viable model, the class of solutions obtained by our approach should satisfy certain regularity and physical requirements . In this section, we analyze the features of our solutions and examine whether the solutions can be used for the description of realistic stars.
Note that we should restrict our solutions only to those values of and for which the energy density , pressure and the electric field intensity remain finite and positive. In addition, and should be so chosen that the gravitational potential remains positive since the other metric function is obviously positive. In (23a) and (23b), we note that and are continuous in the stellar interior. They are also regular at the centre for all values of the parameters and .
That pressure of a realistic star must vanish at a finite boundary implies that
where is given by (22). The above equation puts a restriction on the constants and .
The unique solution to the Einstein-Maxwell system for is given by the Riessner-Nordström line element
For the particular solution, using (37c), we obtain the central density
which implies that . To obtain bounds on other parameters, we evaluate the pressure at two different points. Using equation (37d) at the centre of the star (), we obtain the central pressure
Obviously, we must have
At the boundary of the star (), we impose the condition that the pressure vanishes, i.e., , which yields
Equation (6) determines the radius of the star.
From (37c), we note that the density is always positive and
We must also have
where To fulfill the causality condition , we must have
throughout the interior of the star where