1. Introduction
In the description of astrophysical phenomena, it is of relevance to consider the electromagnetic field, which can be found in certain stellar objects, or spreading through the medium in the form of electromagnetic waves with different frequencies. The stars could contain charge in their interior if their net charge is not zero or as a result of an accretion process. In the case of the exchange between the neutron stars and nearby bodies inside of their own gravitational radius, the difference in potential can be as high as tens of thousands of volts [1]. The possible presence of charge in the interior of the stars has been the focus of different investigation works where they reached the conclusion that the gravitational collapse could be avoided due to the repulsive Coulomb force in the presence of charge [2], a situation that is also present in the case that only considers a charged ball of dust [3, 4]. Also, the presence of the charge results in a greater bound for the rate between the mass M and radius R of the star u = GM/c2R [5–7] than the Buchdahl limit for the chargeless case [8], as such charged models result useful for explaining in an adequate manner the behaviour of massive objects as are Neutron stars, Pulsars and Quark stars. Some works that consider charged models have been applied to the description of compact objects like Crab Pulsar PSR B0531-21, the double pulsar system PSR J0737-3039 [9–12], the SAX J1808.4-3658 star [13], the star PSR J16142230, PSR J1903+327 and LMC X-4 [14, 15].
Also, the effect of the charge has a notable relevance, to such degree that models with a perfect chargeless fluid, which are not physically acceptable as they are, can be electrified and when this occurs the results are models which are physically acceptable, in these cases, the form of the intensity of the electric field is important, an example of this is the interior Schwarzschild solution that does not meet the causality condition, but when it is electrified the causality condition is met, this particular case has the highest number of proposals of electrification [16–23]. Inside of the proposals that take into account the presence of the charge we have different approaches, which can be categorized in two cases, (a) generalizations starting from chargeless solutions to a case in which there is a charge, which are models for which in the absence of charge the chargeless case is recovered and (b) charged models that are not reduced to a chargeless case. Inside of the first class, there have been presented a variety of generalizations from the models of chargeless perfect fluid to the case of a charged perfect fluid, as are the interior solutions of Tolman IV [24, 25], Tolman VI, [24, 26, 27], Tolman VII [24, 28, 29], Wyman–Adler [30–33], Buchdahl [34, 35], Kuchowicz [36, 37], Heintzmann [38, 39], Durgapal (n = 4) [40, 41], Durgapal (n = 5) [40, 42], Vaidya–Tikekar [43–46], Durgapal–Fuloria [47, 48], Knutsen [49, 50], Pant [51–53], Estevez-Delgado [54, 55] among others [13, 56–60]. In the first works, this reduction of a charged model to a chargeless model was an imposed requirement in the construction of charged interior solutions, however, recent investigations focus more on proposals of solutions that satisfy the requirements that the density and pressure are monotonically decreasing functions and that it satisfies the stability criteria [61–63]. Or well in charged models with anisotropic pressures, that is to say, different radial and tangential pressures, for which there is a state equation between the radial pressure and the density Pr = Pr(ρ) the type of state equation that has been analyzed were lineal [64, 65], quadratic [14], Van der Waals [66], Polytropic [67] or Chaplygin type [68]. One alternative which generates a repulsive force effect, without the presence of electric charge, occurs when there is anisotropy in the pressure Δ = Pt − Pr > 0, if Δ < 0 the anisotropy will generate an attraction [69, 70]. In this direction, it has been shown that the compactness is greater than in the chargeless case [71], and that the anisotropy is relevant for obtaining physically acceptable solutions [72]. In the construction of anisotropic solutions geometric properties like the Karmarkar condition have been employed [73–75] The possibility that the interior is formed by core-envelope anisotropic fluid with radial pressure described by a state equation [76] as well as a three-layered, endowed with a distinct equation of states, showing that this type of models can be employed in the description of the strange stars SAX J1808.4-3658, 4U 1820-30, as well as the neutron stars Vela X-1 and PSR J1614-2230 [77]. The relevance of the anisotropy in the pressure and proposals of models with these characteristics could be approached in some other works, however in this work we will focus on the charged case.
Although the requirement that in the absence of charge, the charged solutions are reduced to a physically acceptable chargeless solution is not a demand, even so, the solutions of chargeless perfect fluid are still helpful since they guide the form of the metric potentials that could end up being employed in the description of the interior of the stars, as such they are still relevant when constructing new solutions with perfect fluid and their generalizations. Due to the number of field equations, three, for the Einstein–Maxwell system with perfect fluid, being lower than the number of functions, five, which are the metric coefficients gtt, grr, the density ρ, the pressure P, the charge function q (or the intensity of the electric field E2 = q2/r4), it is possible to solve the equations system by properly assigning a couple of them. Regularly the form of the intensity of the electric field is given and one of the metric functions can be taken as the one analyzed for the case of a perfect fluid, however, there are other alternatives to assign the pair of functions [78]. Given the relevance the presence of the charge has in the interior of the stars, in this investigation report, we present a solution to the Einstein–Maxwell equations with a charged perfect fluid. The solution is a generalization of the chargeless case with metric function [79] ${g}_{{tt}}=-{W}^{2}{\left[5+4{{ar}}^{2}\right]}^{6}/{\left(1+{{ar}}^{2}\right)}^{3}$, where a is a parameter that has units 1/length2 and W is a dimensionless constant whose value will be determined from the continuity condition from the geometry on the surface of the star. In the generalization, we assume a form of the electric field’s intensity that generates a physically acceptable model. The development of the report is as follows: in section 2 we describe the field equations that are solved in section 3 to determine the functions for the density, pressure and intensity of the electric field. In section 4 the set of physical conditions is given and the integration constants are determined. Section 5 is dedicated to the graphic analysis for the star SAX J1808.4-3658, to end with section 6 in which we present the discussion and conclusions of the model presented.
2. The system
Einstein’s equations Gαβ = kTαβ, where k = 8π/Gc4, allows to describe the interior behaviour of compact objects if we consider the energy–momentum tensor:1 ) and the metric (2 ), we obtain the ordinary differential equations system:6 ) is the generalized Tolman–Oppenheimer–Volkoff (TOV) equation [24, 80] for the case of a charged perfect fluid, that can be obtained from the equations (3 )–(5 ), as such from the equations system (3 )–(6 ) only three equations are independent. In our case, we will work with the equations (3 )–(5 ) and the equation (6 ) will be employed to describe the forces present in the interior of the star.
$\begin{eqnarray}\begin{array}{rcl}{T}_{\alpha \beta } & = & ({c}^{2}\rho +{\text{}}P){u}_{\alpha }{u}_{\beta }+{\text{}}{{Pg}}_{\alpha \beta }+\displaystyle \frac{1}{4\pi }[{F}_{\alpha \gamma }{F}_{\beta }{\,}^{\gamma }\\ & & -\displaystyle \frac{1}{4}{F}^{\sigma \gamma }{F}_{\sigma \gamma }{g}_{\alpha \beta }],\end{array}\end{eqnarray}$
where P represent the pressure, ρ is the density of the matter measured by an observer with 4-velocity, uμ; while gαβ denotes the components of the static and spherically symmetric spacetime described by: $\begin{eqnarray}\begin{array}{rcl}{\rm{d}}{s}^{2} & = & -y{\left(r\right)}^{2}\,{\rm{d}}{t}^{2}+\displaystyle \frac{{\rm{d}}{r}^{2}}{B(r)}\\ & & +{r}^{2}({\rm{d}}{\theta }^{2}+{\sin }^{2}\theta {\rm{d}}{\phi }^{2}).\end{array}\end{eqnarray}$
From Einstein’s equations and the conservation equation ∇μTμ ν = 0, for the momentum–energy tensor ( $\begin{eqnarray}{{kc}}^{2}\rho +{E}^{2}=-\displaystyle \frac{B^{\prime} }{r}+\displaystyle \frac{1-B}{{r}^{2}}\end{eqnarray}$
$\begin{eqnarray}{kP}-{E}^{2}=\displaystyle \frac{2{By}^{\prime} }{{ry}}-\displaystyle \frac{1-B}{{r}^{2}}\end{eqnarray}$
$\begin{eqnarray}{kP}+{E}^{2}=\displaystyle \frac{{By}^{\prime\prime} }{y}+\left(\displaystyle \frac{B^{\prime} }{2}+\displaystyle \frac{B}{r}\right)\displaystyle \frac{y^{\prime} }{y}+\displaystyle \frac{B^{\prime} }{2r}\end{eqnarray}$
$\begin{eqnarray}{\text{}}P^{\prime} =-\left({\text{}}P+{c}^{2}\rho \right)\displaystyle \frac{y^{\prime} }{y}+2\displaystyle \frac{E}{{r}^{2}}[{r}^{2}E]^{\prime} ,\end{eqnarray}$
here ′ denotes the derivative of the function in regard to r, ″the second derivative, P is the pressure in the interior of the star, E2 = q2(r)/r4 represents the intensity of the electric field, q(r) the electric charge: $\begin{eqnarray*}q(r)=4\pi {\int }_{0}^{r}\displaystyle \frac{\sigma {r}^{2}}{\sqrt{B}}={r}^{2}\sqrt{-{F}^{{rt}}{F}_{{rt}}}={r}^{2}{F}^{{rt}}\displaystyle \frac{y}{\sqrt{B}},\qquad \end{eqnarray*}$
in the interior of the radius sphere r and σ represents the density of the charge given by $\begin{eqnarray}\sigma =\displaystyle \frac{\sqrt{B}[{r}^{2}E]^{\prime} }{4\pi {r}^{2}}.\end{eqnarray}$
The equation (3. The solution
There are several ways in which we can obtain a solution to Einstein’s equations with a charged perfect fluid. One of them which is important, and that allows to generalize an existing physically acceptable solution for the chargeless case to the charged case, consists in taking one of the metric potentials gtt or grr which corresponds to a solution with perfect fluid and solve the equations system (3 )–(5 ) assuming an appropriate form for the electric field’s intensity [81–87]. And although, by construction, the solution is physically acceptable for the case in which E(r) = 0, there is no guarantee that the new solution with electric field will be physically acceptable and this is what makes it a difficult task to realize. We will begin from the metric potential ${g}_{{tt}}=-y{\left(r\right)}^{2}$, with3 )–(5 ), through the mechanism that is developed in this work, requires the proposal of an explicit form of the electric field’s intensity E. The proposal of the functional form of the intensity was chosen in a way that not only facilitates the solution of the system but that it also satisfies with physical characteristics required for describing the intensity of the electric field in the interior of the star, specifically, it must be met that E(0) = 0, besides that in a region r ∈ (0, b), E(r) must be a monotonic increasing function. Taking into account these requirements we will assume an electric field’s intensity:4 ) and (5 ) and replacing the form of the functions given by (8 ) and (9 ) it results in the differential equation10 ) which will be determined in terms of the parameter a and the radius of the star when imposing that the pressure on the surface is zero. With the obtaining of the function B(r) or equivalently the metric component grr = 1/B, the geometry is determined in a general manner. To represent a compact object with values of mass and radius from an appropriate form, it is necessary to impose conditions on the behaviour of the geometry and from the functions of the density and pressure. That, for the solution we present, once we replace the form of the metric functions and in the electric field we reach:
$\begin{eqnarray}y\left(r\right)=W{\left(\displaystyle \frac{5+4\,{{ar}}^{2}}{\sqrt{1+{{ar}}^{2}}}\right)}^{3},\end{eqnarray}$
which was employed to obtain a solution with perfect fluid and that was used to represent the interior of the star PSR J0348+0432. The integration of the coupled ordinary differential equations system ( $\begin{eqnarray}E{\left(r\right)}^{2}=\displaystyle \frac{64\nu \left(1+2\,{{ar}}^{2}\right){\left(1+{{ar}}^{2}\right)}^{4}{a}^{2}{r}^{2}}{{\left(5+8\,{{ar}}^{2}\right)}^{3}{\left(5+4\,{{ar}}^{2}\right)}^{5}},\end{eqnarray}$
with ν ≥ 0 a parameter associated with the charge, that for ν = 0 allows to recover the chargeless case. Subtracting the equations ( $\begin{eqnarray}\begin{array}{l}B^{\prime} -\displaystyle \frac{2(25+90{{ar}}^{2}+82{a}^{2}{r}^{4}-32{a}^{4}{r}^{8})B}{(1+{{ar}}^{2})(5+4{{ar}}^{2})(1+2{{ar}}^{2})(5+8{{ar}}^{2})r}\\ \quad +\displaystyle \frac{2(1+{{ar}}^{2})(5+4{{ar}}^{2})}{(1+2{{ar}}^{2})(5+8{{ar}}^{2})r}\\ \quad +\displaystyle \frac{256{\left(1+{{ar}}^{2}\right)}^{5}{a}^{2}\nu \,{r}^{3}}{{\left(5+8\,{{ar}}^{2}\right)}^{4}{\left(5+4\,{{ar}}^{2}\right)}^{4}}=0,\end{array}\end{eqnarray}$
solving, we obtain the function: $\begin{eqnarray}\begin{array}{l}B(r)=-\displaystyle \frac{{\left(1+{{ar}}^{2}\right)}^{2}[S(r)+62528{a}^{4}{r}^{8}+65536{a}^{5}{r}^{10}]}{{\left(5+8{{ar}}^{2}\right)}^{3}{\left(5+4{{ar}}^{2}\right)}^{4}}\\ \quad +\displaystyle \frac{64(1+2{{ar}}^{2}){\left(1+{{ar}}^{2}\right)}^{5}{{ar}}^{2}}{3{\left(5+8{{ar}}^{2}\right)}^{7/2}{\left(5+4{{ar}}^{2}\right)}^{4}}\\ \quad \times \left[3\left(1377-\nu \right)\mathrm{arctanh}\left[\displaystyle \frac{\sqrt{5+8{{ar}}^{2}}}{3+4{{ar}}^{2}}\right]\right.\\ \quad \left.-191\sqrt{3}\arctan \left[\displaystyle \frac{\sqrt{3}\sqrt{5+8{{ar}}^{2}}}{1+4{{ar}}^{2}}\right]+C\right],\end{array}\end{eqnarray}$
where S(r) = 78125 + 367138ar2 + 568536a2r4 + 282528a3r6 and C is the constant of integration that originates when solving the equation ( $\begin{eqnarray}\begin{array}{lcl}{{kc}}^{2}\rho (r) & = & \displaystyle \frac{3(25+105{{ar}}^{2}+142{a}^{2}{r}^{4}+72{a}^{3}{r}^{6})(1-B)}{(1+2{{ar}}^{2})(5+8{{ar}}^{2})(5+4{{ar}}^{2})(1+{{ar}}^{2}){r}^{2}}\\ & & -\displaystyle \frac{64{\left(1+{{ar}}^{2}\right)}^{4}{W}_{0}(r)\nu {a}^{2}{r}^{2}}{{\left(5+4{{ar}}^{2}\right)}^{5}{\left(5+8{{ar}}^{2}\right)}^{4}}\\ & & +\displaystyle \frac{6{a}^{2}{r}^{2}\left(13+24\,{{ar}}^{2}+16\,{a}^{2}{r}^{4}\right)}{\left(1+2\,{{ar}}^{2}\right)\left(5+8{{ar}}^{2}\right)\left(5+4{{ar}}^{2}\right)\left(1+{{ar}}^{2}\right)},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{kP}\left(r\right) & = & \displaystyle \frac{6a\left(3+4\,{{ar}}^{2}\right)}{\left(5+4\,{{ar}}^{2}\right)\left(1+{{ar}}^{2}\right)}\\ & & -\displaystyle \frac{\left(5+7\,{{ar}}^{2}\right)\left(1+4\,{{ar}}^{2}\right)\left(1-B\right)}{\left(5+4\,{{ar}}^{2}\right){r}^{2}\left(1+{{ar}}^{2}\right)}\\ & & +\displaystyle \frac{64\left(1+2\,{{ar}}^{2}\right){\left(1+{{ar}}^{2}\right)}^{4}{r}^{2}\nu \,{a}^{2}}{{\left(5+4\,{{ar}}^{2}\right)}^{5}{\left(5+8\,{{ar}}^{2}\right)}^{3}},\end{array}\end{eqnarray}$
where W0(r) = 25 + 54ar2 + 32a2r4. The conditions that are imposed on the hydrostatic functions and the geometry allow us to set the constants in terms of the compactness and the charge-radius rate for the star, which likewise determine the type of objects that can be represented with the obtained solution.4. Physical conditions
The conditions which are necessary to satisfy for the solution to be physically acceptable can be classified into two groups, those that must satisfy the hydrostatic functions (ρ, P) and the intensity of the electric field E2 and those that must satisfy the geometry (y, B). Although two of these are equivalent, specifically, that the pressure is nullified on the surface and the continuity of the second fundamental form of the exterior and interior metric lead to the same relation between the integration constants, the rest of the conditions are complementary. In the first group of conditions, we have that:
(a1) The density and the pressure must be regular, positive and monotonically decreasing functions and the pressure must be zero on the surface, identified by r = R.
(a2) The intensity of the electric field must be regular, positive, monotonically increasing and it must be zero in the center of the star and on the surface of the star $E{\left(R\right)}^{2}\,=\,{Q}^{2}/{R}^{4}$, where Q is the total charge of the star.
(a3) The causality condition must not be violated, that is to say, the speed of sound must be positive and lower than the speed of light in the vacuum
$\begin{eqnarray*}0\leqslant {v}^{2}=\displaystyle \frac{\partial P(\rho )}{\partial \rho }\leqslant {c}^{2}.\end{eqnarray*}$
(a4) The following energy conditions must be satisfied: null energy condition (NEC), weak energy condition (WEC), strong energy condition (SEC) and the dominant energy condition (DEC) which, in summary, can be met if the following equations are satisfied:
$\begin{eqnarray}\begin{array}{l}{c}^{2}\rho +P\geqslant 0,\,{{kc}}^{2}\rho +{kP}+2{E}^{2}\geqslant 0,\\ {{kc}}^{2}\rho +3{kP}+2{E}^{2}\geqslant 0,\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{c}^{2}\rho -P\geqslant 0,\,{{kc}}^{2}\rho -{kP}+2{E}^{2}\geqslant 0,\\ {{kc}}^{2}\rho +{E}^{2}\geqslant 0.\quad \end{array}\end{eqnarray}$
(a5) The solution must be stable in regards to infinitesimal radial adiabatic perturbation, that is to say, the adiabatic index [38, 88–91]
$\begin{eqnarray}\gamma =\frac{{c}^{2}\rho +P}{P}\frac{\partial P(\rho )}{\partial \rho }\gt {\gamma }_{\mathrm{crit}}=\frac{4}{3}+\frac{19}{21}u.\end{eqnarray}$
And in a complementary manner the criteria of Harrison–Zeldovich–Novikov must be satisfied, that is to say, $\tfrac{\partial M}{\partial {\rho }_{c}}\gt 0$ with ρc the central density. Which guarantees the stability of the gaseous stellar configuration in relation to radial pulsations.The conditions that the geometry must satisfy are:
(b1) The geometry must be regular in the interior of the star and it must be absent of event horizons.
(b2) The interior and exterior geometries (described by the Reissner Nordstrom metric):
$\begin{eqnarray}\begin{array}{rcl}{\rm{d}}{s}^{2} & = & -\left[1-\displaystyle \frac{2{GM}}{{c}^{2}r}+\displaystyle \frac{{Q}^{2}}{{r}^{2}}\right]{\rm{d}}{t}^{2}+{\left[1-\displaystyle \frac{2{GM}}{{c}^{2}r}+\displaystyle \frac{{Q}^{2}}{{r}^{2}}\right]}^{-1}{\rm{d}}{r}^{2}\\ & & +{r}^{2}{\rm{d}}{{\rm{\Omega }}}^{2},\quad r\geqslant R,\end{array}\end{eqnarray}$
as well as the second fundamental form, must be continual on the surface.From this last condition we obtain the constants a, ν, C and W
$\begin{eqnarray}a=\displaystyle \frac{27\,u-18\,{q}^{2}-9+\sqrt{81-246u+169{u}^{2}+4\left(21-23u\right){q}^{2}+4\,{q}^{4}}}{8\left(3-7\,u+4\,{q}^{2}\right){R}^{2}},\end{eqnarray}$
$\begin{eqnarray}\nu =\displaystyle \frac{{\left(5+8\,{{aR}}^{2}\right)}^{3}{\left(5+4\,{{aR}}^{2}\right)}^{5}{q}^{2}}{64\,{a}^{2}\left(1+2\,{{aR}}^{2}\right){\left(1+{{aR}}^{2}\right)}^{4}{R}^{4}},\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}C & = & 191\sqrt{3}\arctan \left[\displaystyle \frac{\sqrt{3}\sqrt{5+8\,{{aR}}^{2}}}{1+4\,{{aR}}^{2}}\right]\\ & & +3\left(\nu -1377\right)\mathrm{arctanh}\left[\displaystyle \frac{\sqrt{5+8\,{{aR}}^{2}}}{3+4\,{{aR}}^{2}}\right]\\ & & -\displaystyle \frac{3\sqrt{5+8{{aR}}^{2}}\left(N+898190+5887971\,{{aR}}^{2}\right)}{64(5+7{{aR}}^{2})(1+2{{aR}}^{2})(1+4{{aR}}^{2}){\left(1+{{aR}}^{2}\right)}^{4}}\\ & & -\displaystyle \frac{3\sqrt{5+8{{aR}}^{2}}\nu {\text{}}{{aR}}^{2}}{(5+7{{aR}}^{2})(1+4{{aR}}^{2})},\end{array}\end{eqnarray}$
$\begin{eqnarray}{W}^{2}=\displaystyle \frac{\left(1-2u+{q}^{2}\right){\left(1+{{aR}}^{2}\right)}^{3}}{{\left(5+4\,{{aR}}^{2}\right)}^{6}},\end{eqnarray}$
where N = 14277600 a3R6 − 438528 a4R8 − 12575808 a5R10 − 9615104 a6R12 − 2359296 a7R14 + 14385882 a2R4. These relations show that the only parameters that determine the solution are q = Q/R, u = GM/c2R and aR. Besides these relations, we have a set of inequalities that arise from imposing the rest of the requirements and these are the ones that determine the admissible intervals for u and q.5. Graphic analysis
A detailed analysis allows us to observe that for values of compactness such as u ≈ 0.433 the intensity of the electric field stops being a monotonically decreasing function near the surface of the star and although this behaviour could be explained as a result of the decrease in the excess of positive and negative charges in the surface of the star, in this report we will assume that the condition a2 is satisfied, that is to say, that the intensity of the electric field is a monotonically increasing function. For the graphic representation, we define the dimensionless variable as x = r/R, in such case x ∈ [0, 1], locating the surface of the star in x = 1 and the dimensionless functions kc2R2ρ(x), kR2P(x), ${R}^{2}E{\left(x\right)}^{2}$, $v{\left(x\right)}^{2}/{c}^{2}$ and kR3Fi(x), where Fi represents the gravitational, electric and hydrostatic forces. In addition to these functions, we will present the graphs of the adiabatic index γ(x) and the metric functions −gtt(x) and 1/grr(x).
5.1. Application of the solution for the star SAX J1808.4-3658
The graphic analysis of the solution will focus on the star SAX J1808.4-3658 (SS1) with mass M = 1.435M⊙ and radius R = 7.07 km which generates a compactness u = 0.2997 [92], and for this type of stars with a high compactness, it is appropriate to realize a representation of their interior either considering that the star presents anisotropic pressures or that the star is charged and it is precisely the presence of the charge what would generate the anisotropy in the effective radial and effective tangential pressures. From the set of requirements for the solution to be physically acceptable, is the positivity of the pressure and its monotonically decreasing behaviour that limits the maximum value of the charge radius parameter q ≤ 0.293 50. In figure 1 it is shown the regular, monotonically decreasing and positive behaviour of the pressure, it is also noted that when q → 0.293 50 the pressure in the vicinity of the surface of the star approaches zero. If we consider values q > 0.293 50 the value of P < 0, as such the admissible values of q that allow a physically acceptable behaviour of the pressure for a compactness u = 2997 are for q ∈ [0, 0.29 350]. For this range, the rest of the required conditions are satisfied without the need for any additional restrictions. Also, figure 1 shows that the presence of the charge implies that as the value increases (as seen by the parameter q) the hydrostatic pressure diminishes, which is consistent with that expected by the repulsion effect between the charges, with the maximum value of the pressure occurring when the net charge in the interior is zero (q = 0). In figure 2 it is shown that the density is a monotonically decreasing, regular and positive function. The charge in this case makes it so that the density is greater in the center of the star. From figure 3 we have that the speed of sound can be a monotonically increasing or decreasing function and the most relevant part is that the causality condition is not violated, that is to say, the speed of sound v2 is positive and lower than the speed of light (0 < v2 < c2). The intensity of the electric field is shown in figure 4, the graph for each one of the charge parameters q are monotonically increasing functions, positive, bounded, regular and zero on the center of the star. Its value on the surface of the star increases as the charge parameter increases.
Figure 1. Graphs of the pressure for some admissible values of the electric charge parameter. |
Figure 2. Behaviour of the density for different values of the charge parameter. |
Figure 3. The tangential speed of sound. |
Figure 4. Intensity of the electric field for different charge parameters. |
Although, as a result of the continuity of the metric and from the second fundamental form we obtained the constants in terms of the compactness parameters u and charge radius rate q, it is illustrative to show the continuity of the metric functions on the surface of the star (x = 1). In figure 5 we represent the metric functions in the interior x ≤ 1 and the metric functions in the exterior generated by the Reissner–Nordstrom metric (x ≥ 1)
Figure 5. Continuity of the metric coefficients that determine the interior geometry ($-{g}_{{tt}}^{(i)}$, $1/{g}_{{rr}}^{(i)}$ ) and the exterior geometry ($-{g}_{{tt}}^{(e)}=1/{g}_{{rr}}^{(e)}$ ). |
5.2. Analysis of stability and equilibrium
For the analysis of the stability we begin applying the Harrison–Zeldovich–Novikov criteria which guarantees the stability of the gaseous stellar configuration in relation to radial pulsations if $\tfrac{\partial M}{\partial {\rho }_{c}}\gt 0$, where ρc represents the central density. The function of mass for the model that is being analyzed is obtained considering the functions (12 ) and (13 ) evaluated in r = 0 and taking into account the form of the constant (18 ), resulting:
$\begin{eqnarray}M({\rho }_{c},Q)=\displaystyle \frac{{c}^{2}[1458{Q}^{2}+243({c}^{2}{\rho }_{c}+3{P}_{c}){{kR}}^{2}{s}_{1}+10{\left({c}^{2}{\rho }_{c}+3{P}_{c}\right)}^{2}{k}^{2}{R}^{4}{s}_{2}]}{G(70{\left({c}^{2}{\rho }_{c}+3{P}_{c}\right)}^{2}{k}^{2}{R}^{4}+729({c}^{2}{\rho }_{c}+3{P}_{c}){{kR}}^{2}+1458)R},\end{eqnarray}$
where s1 = 2Q2 + R2 and s2 = 4Q2 + 3R2. The partial derivative of this function in regards to the central density is: $\begin{eqnarray}\displaystyle \frac{\partial M}{\partial {\rho }_{c}}=\displaystyle \frac{486{{kRc}}^{4}[729+180({c}^{2}{\rho }_{c}+3{P}_{c}){{kR}}^{2}+10{\left({c}^{2}{\rho }_{c}+3{P}_{c}\right)}^{2}{k}^{2}{R}^{4}]{s}_{3}}{{\left[54+7k\left({c}^{2}{\rho }_{c}+3\,{P}_{c}\right){R}^{2}\right]}^{2}{\left[27+10\,k\left({c}^{2}{\rho }_{c}+3\,{P}_{c}\right){R}^{2}\right]}^{2}G},\end{eqnarray}$
with s3 = R2 − Q2. The Harrison–Zeldovich–Novikov criteria being satisfied if R2 − Q2 > 0 since the rest of the terms are positive. On the other hand, the graphs of figure 6 in which the adiabatic index is represented imply that the solution is stable in relation to infinitesimal radial adiabatic perturbation, since it is satisfied that γ(x) > γcrit ≈ 1.6044, for each one of the values of the charge parameter q [38, 88–91].
Figure 6. Behaviour of the adiabatic index for the compactness u = 0.2997 with critical adiabatic index γcrit ≈ 1.6044. |
The equilibrium in the interior of the star, according to the generalized TOV equation for the charged case is:25 ), its approximated moment of inertia is I = 6.8637 × 1038 kg m2.
$\begin{eqnarray*}\begin{array}{l}-\displaystyle \frac{\left({\text{}}P+{c}^{2}\rho \right)y^{\prime} }{y}-{\text{}}{P}_{r}^{\prime} +\displaystyle \frac{1}{{r}^{4}}\left[{r}^{4}E{\left(r\right)}^{2}\right]^{\prime} \\ \quad ={F}_{g}+{F}_{h}+{F}_{e}=0,\end{array}\end{eqnarray*}$
which indicates that it is maintained as a result of the gravitational attractive force Fg and the forces of hydrostatic Fh and electric Fe repulsion, given by $\begin{eqnarray}\begin{array}{rcl}{F}_{g}(r) & = & -\displaystyle \frac{\left({\text{}}{P}_{r}+{c}^{2}\rho \right)y^{\prime} }{y}\qquad {F}_{h}(r)=-{\text{}}{P}_{r}^{\prime} \\ {F}_{e}(r) & = & \displaystyle \frac{1}{{r}^{4}}\left[{r}^{4}E{\left(r\right)}^{2}\right]^{\prime} .\end{array}\end{eqnarray}$
Each one of these forces are shown in figure 7, the long-dash lines correspond to the graphs of the gravitational force, the electric force is represented by the solid lines and the dash lines describe the behaviour of the force associated to the gradient of the pressure. The repulsion force which has the greatest contribution when the charge parameter is small occurs due to the pressure gradient and as the charge parameter increases the electric force increases (black colored lines), which shows the relevance of the presence of the charge in the model. Another of the characteristics that defines a compact object is its moment of inertia, which for the case in which RΩ/c ≪ 1, a good approximation implies that the moment of inertia is given by [77, 93] $\begin{eqnarray}I=\displaystyle \frac{2}{5}\left[1+\displaystyle \frac{M\,\mathrm{km}}{R\,{M}_{\odot }}\right]{{MR}}^{2}.\end{eqnarray}$
In the case of the star SAX J1808.4-3658 it is satisfied that RΩ/c = 0.059 38 ≪ 1 [94], as such, it can be considered as a slowly rotating object. So, in accordance with the relation (
Figure 7. Forces that intervene in the stellar equilibrium for the charged and chargeless case. |
6. Discussion and conclusions
Although the graphic behaviour shows that all the requirements for the solution to be physically acceptable are satisfied and that, as well as it being consistent, as a result of the electric repulsion effect, the hydrostatic pressure diminishes when the charge increases, it is convenient to obtain the physical values of the density in the surface and in the center of the star as well as the central pressure and speed of sound, to know if the orders of magnitude are also consistent. From the table 1 we have that the values of the density in the center and on the surface, as well as the pressure are characteristic of compact stars. In addition, we also report the values of the net charge in the interior of the star being its maximum value of 2.4085 × 1020C which corresponds to the typical values obtained with other models for compact stars.
Table 1. Physical values of the central density ρc, on the surface ρb, central pressure Pc, speed of sound and central adiabatic index for the star SAX J1808.4-3658. |
| q | Q | ρc | ρb | Pc | vc2 | vb2 | γc |
|---|---|---|---|---|---|---|---|
| 1020C | ${10}^{18}\tfrac{\mathrm{kg}}{{{\rm{m}}}^{3}}$ | ${10}^{18}\tfrac{\mathrm{kg}}{{{\rm{m}}}^{3}}$ | 1034Pa | c2 | c2 | ||
| 0 | 0 | 2.9069 | 1.4839 | 9.7066 | 0.8383 | 0.6839 | 3.0943 |
| 0.07337 | 0.6021 | 2.9232 | 1.4589 | 8.9223 | 0.6974 | 0.6417 | 2.7507 |
| 0.14674 | 1.2042 | 2.9610 | 1.3809 | 6.7939 | 0.4347 | 0.5091 | 2.1376 |
| 0.22012 | 1.8063 | 2.9919 | 1.2408 | 3.8567 | 0.2150 | 0.2819 | 1.7139 |
| 0.29350 | 2.4085 | 2.9979 | 1.0239 | 0.6567 | 0.0565 | 0.0037 | 2.3754 |
In conclusion, we have proposed a new charged stellar solution in the context of Einstein’s general relativity theory, and realized its analysis. The model considers a static and spherically symmetric spacetime that contains a charged perfect fluid. In the absence charge we recover the chargeless case presented recently [79], differing from this last one because its compactness value is greater due to the presence of the charge. Meanwhile from the analysis of the hydrostatic equilibrium equation, a generalization of the TOV equation for a charged case, it can be seen that the electric force is able to contribute more in the repulsion that the pressure gradient. The solution is stable both in regards to the Harrison–Zeldovich–Novikov criteria as well as in relation to the adiabatic index criteria. Also, the orders of magnitude that the model generates for the hydrostatic functions are consistent with the values expected for this type of star. The solution presented leaves still an open question associated with the behaviour that the speed of sound has in the presence of the charge, since it can be monotonically increasing or monotonically decreasing. This leads us to the question of what is the mechanism that generates this behaviour, a question that could be approached in future works.