A family of solutions to the Einstein-Maxwell system of equations describing relativistic charged fluid spheres

A family of solutions to the Einstein-Maxwell system of equations describing relativistic charged fluid spheres

K. Komathiraj1    Ranjan Sharma2

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.

Relativistic star, Exact solutions, Einstein-Maxwell system.

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.



1 Introduction

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 [1], Majumdar [2], Bonner [3, 4], Stettner [5], Bekenstein [6] and Cooperstock and Cruz [7]. A detailed review of exact solutions to Einstein-Maxwell systems and their physical acceptability can be found in the compilation work of Ivanov [8]. 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 [21] and Komathiraj and Maharaj [22] have obtained solutions for charged strange quark stars admitting a linear equation of state (EOS). Thirukkanesh and Maharaj [23] have analyzed the role of anisotropy on the physical behaviour of a given charged distribution admitting a linear EOS. Varela et al [24] have analyzed features of a charged anisotropic fluid distribution admitting linear as well as non-linear EOS. Takisa and Maharaj [25] have obtained a new class of solutions for a charged quark matter distribution. Feroze and Siddiqui [26] and Maharaj and Takisa [27] 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 [30] have presented a new family of exact solutions to the Einstein-Maxwell system for an anisotropic charged matter on the Finch and Skea [31] background spacetime. A class of charged anisotropic stellar solutions has been developed and studied by Murad and Fatema [32]. Hansraj et al [33] have analyzed all static-charged dust sphere models in general relativity. Recently, Sunzu and Danford [34] 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 [35] which was, in fact, a generalization of the superdense stellar model developed by Tikekar [36]. 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


proposed by Durgapal and Bannerji [37], where and are constants. Under the transformation (3), the system (2) becomes


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 [38] charged stellar model which is a generalization of the stellar models developed previously by Finch and Skea [31] and Hansraj and Maharaj [39]. 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.

Substitution of (5) in (4c) yields


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 [40] and John and Maharaj [41] and can also be reduced to the uncharged stellar model developed by Maharaj and Mkhwanazi [42]. Substituting (9) in equation (8), we obtain


which is the master equation for the system of equations (4). For , the differential equation (10) reduces to


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 [43] 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.

For a legitimate solution, we need to determine the coefficients as well as the parameter . Substituting (17) in the differential equation (15), we obtain the indicial equation

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.

We generate two linearly independent solutions to (15) with the help of (17) and (19). For the parameter value , we obtain the first solution

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



Thus the general solution to the differential equation (3), for the choice of the electric field (9), is given by


where and are arbitrary constants. Using, (4) and (22), we write the exact solution to the Einstein-Maxwell system in the form


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


Using (17) (when ) and (25), we obtain


where .

On substituting in (18) and by setting , we obtain


where is a fixed integer. Obviously, and the subsequent coefficients vanish. Equation (27) yields


Using (17) (when ) and (28), we obtain


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 and . Using(29), the second category of general solutions to (15) is obtained as


where . In (5.2) - (5.2), and are integration constants. In terms of the original variable , it is possible to write (5.2) 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.

We are now in a position to integrate equations (5.2) and (5.2) for specific values of the parameters and .

For , equation (5.2) becomes


where . Also for , equation (5.2) takes the form


where .

By setting and in (34), we obtain

where we have assumed and . Thus, we have regained the Durgapal and Bannerji solution [37].

If we set and in (35), we obtain

where we have assumed and . This class of solutions was found earlier by Maharaj and Mikhwanazi [42].

Further, by setting and in (35), we obtain

where we have assumed and . This particular solution corresponds to the charged stellar model of John and Maharaj [41]. Note that a minor error appearing in the John and Maharaj paper [41] has been addressed in this work.

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)
Case I:
In this case (34) becomes


where we have set

Subsequently, the general solution to the Einstein-Maxwell system (23) can be expressed as


where and

Case II:
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 [44]. 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


where and are the total mass and the charge of the star. Matching of the line elements (1) and (39), across the boundary , yields the relationships between the constants and as follows:

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


as .
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