1. Introduction
Objects with a greater compactness than in the case of a perfect fluid, corresponding to the Buchdahl limit [1] u ≡ GM/c2R < 4/9, can occur when we have a difference between the radial pressure Pr and the tangential pressure Pt, known as anisotropy Δ = Pt − Pr [2], in the presence of electric charge [3–5] or when we have both situations. The anisotropy in relation to compact objects has been addressed in the analysis of stars with ordinary matter for an incompressible fluid [6] and with non-constant density [7–21], in the analysis of bosonic stars [22], in the context of Brans–Dicke gravity [23], in alternative gravitation theories like the f(T) [24], in the consideration of stars with quintessence type matter [25–29] and also in the analysis of hypothetical objects such as gravastars [30]. In the charged case we have models that consider a perfect charged fluid [31–37] as well as in the case of objects with charged anisotropic fluid [38–46]. From a theoretical point of view, the anisotropy may occur for different motives: when the density of the matter is superior to 1015g/cm3 Δ ≠ 0 as a result that at these orders the interactions are relativistic [47]; phase transitions of the matter, when changing to a superfluid and superconductive state, that can occur between the internal core and the crust [48, 49]; in addition to these causes we have that the presence of electric charges facilitates the generation of strong magnetic fields and these can also produce anisotropy [50, 51]. In the conditions which are believed to exist in the interior of the stars, many of the proposals are only justified by the theoretical development in areas such as particle and nuclear physics. Starting from these we assume that the compact stars are not only formed by neutrons but by a mixture of hadronic matter and quarks [52] or by exotic matter such as hyperons, Kaon condensates or a deconfined phase of strange matter [53]. In the case of compact stars in which we consider the presence of quarks, a state equation that adequately describes the interaction is the so-called MIT Bag equation, given by Pr = (c2ρ − 4Bg)/3 associated with the interaction of the strong nuclear force. However it is not the only component present in the interior of the stars, the presence of the electric charge may facilitate the existence of a strong magnetic field that generates the anisotropy, as such, considering this possibility, it is convenient to analyse models that consider the presence of electric charge, a situation that will be analysed in this work. We will consider a static and spherically symmetric spacetime, with matter described by a charged anisotropic fluid and with radial pressure described by the MIT Bag equation [54]. As a complementary tool in the solution of the equations from the Einstein–Maxwell system, we will assume that the geometry of the interior satisfies the Karmarkar condition, which relates, through a second order differential equation the metric components gtt and grr. Said equation guarantees that the four geometry that describes the interior of the star can be immersed in a dimension 5 pseudo-Euclidean manifold. Although we have been unable to clearly comprehend the why of such a condition and as a consequence of this, the equation allows us to generate physically acceptable solutions, and this has been employed in a variety of works [55–67]. In one of the more recent works, it has been proposed that for the solution of this equation a new form of the metric potential ${g}_{{rr}}={\left[1+{{Car}}^{2}{\arctan }^{2}\left(d+{{ar}}^{2}\right)\right]}^{-1}$ and from this choice, without assuming a state equation, there have been solutions constructed which are consistent with the observational data of the stars EXO 1785-248 and SMC X-4. Motivated by these works, in this report we will assume the same form of the metric potential grr with the difference that here we assume a MIT Bag state equation and we will show the consistency of our proposal with the observational data that generates the greater compactness, that is to say with mass M = 1M⊙ and radius R = 7.69 km for the star Her X-1.
The paper is organized as follows. In section 2 we present the field equations and we address the Karmarkar condition. In section 3 we present the solution and we mention the requirements for it to be physically acceptable. Section 4 is centered on the graphic analysis and application of the solution for the description of the star Her X-1 showing the stability of the solution. In section 5 we mention the conclusions and we discuss future works as a result of the geometry proposed.
2. The system
The equations that describe the interior behaviour of charged objects with densities greater than that of the nuclear density are given by Einstein's equations, Gαβ = kTαβ, k = 8π/Gc4, with the energy-momentum tensor described by:6 ) is the result of imposing the Bianchi identity which leads us to the equation of conservation ∇μTμ ν = 0 which generalizes the Tolman–Oppenheimer–Volkoff (TOV) equation [68, 69] and although this is satisfied if the equations (3 )–(5 ) are met and as such might not be included in the equation system, it is convenient in the analysis of the forces present in the interior of the stars.
$\begin{eqnarray}\begin{array}{rcl}{T}_{\alpha \beta } & = & ({c}^{2}\rho +{\text{}}{P}_{t}){u}_{\alpha }{u}_{\beta }+{\text{}}{P}_{t}{g}_{\alpha \beta }+({\text{}}{P}_{r}-{P}_{t}){\chi }_{\alpha }{\chi }_{\beta }\\ & & +\displaystyle \frac{1}{4\pi }[{F}_{\alpha \gamma }{F}_{\beta }{\,}^{\gamma }-\displaystyle \frac{1}{4}{F}^{\sigma \gamma }{F}_{\sigma \gamma }{g}_{\alpha \beta }],\end{array}\end{eqnarray}$
where χμ is the unit vector in the radial direction, Pr and Pt represent the radial and tangential pressures respectively, while ρ is the density of the matter measured by an observer with 4-velocity and uμ and gαβ denote the components of the static and spherically symmetric spacetime described by the line element: $\begin{eqnarray}{\mathrm{ds}}^{2}=-{{\rm{e}}}^{2\nu (r)}{\mathrm{dt}}^{2}+{{\rm{e}}}^{\lambda (r)}{\mathrm{dr}}^{2}+{r}^{2}({\rm{d}}{\theta }^{2}+{\sin }^{2}\theta {\rm{d}}{\phi }^{2}).\end{eqnarray}$
The non-zero components of Einstein's equations result in the ordinary differential equations system: $\begin{eqnarray}{{kc}}^{2}\rho +{E}^{2}=\displaystyle \frac{{{\rm{e}}}^{-\lambda }}{r}\lambda ^{\prime} +\displaystyle \frac{1-{{\rm{e}}}^{-\lambda }}{{r}^{2}},\end{eqnarray}$
$\begin{eqnarray}{{kP}}_{r}-{E}^{2}=\displaystyle \frac{2{{\rm{e}}}^{-\lambda }}{r}\nu ^{\prime} -\displaystyle \frac{1-{{\rm{e}}}^{-\lambda }}{{r}^{2}},\end{eqnarray}$
$\begin{eqnarray}{{kP}}_{t}+{E}^{2}=\left[\nu ^{\prime\prime} +\displaystyle \frac{1}{2r}\left(2\nu ^{\prime} -\lambda ^{\prime} \right)\left(1+\nu ^{\prime} \right)\right]{{\rm{e}}}^{-\lambda },\end{eqnarray}$
$\begin{eqnarray}{P}_{r}^{{\prime} }=-\left({\text{}}{P}_{r}+{c}^{2}\rho \right)\nu ^{\prime} +\displaystyle \frac{2}{r}({P}_{t}-{P}_{r})+2\displaystyle \frac{E}{{r}^{2}}[{r}^{2}E]^{\prime} ,\end{eqnarray}$
where E2 = q2(r)/r4 represents the intensity of the electric field $\begin{eqnarray*}q(r)=4\pi {\int }_{0}^{r}\sigma {r}^{2}{{\rm{e}}}^{\lambda /2}{\rm{d}}r={r}^{2}\sqrt{{F}^{{rt}}{F}^{{rt}}}={r}^{2}{F}^{{rt}}{{\rm{e}}}^{\nu +\tfrac{\lambda }{2}},\qquad \end{eqnarray*}$
is the charge in the interior of the radius sphere r and σ represents the density of the charge given by $\begin{eqnarray}\sigma =\displaystyle \frac{[{r}^{2}E]^{\prime} }{4\pi {r}^{2}{{\rm{e}}}^{\lambda /2}}\end{eqnarray}$
and ′ denotes the derivative with respect to the radial coordinate. Equation (A tool that is very useful in obtaining the solutions of Einstein's equations both for the chargeless and charged case when we have a perfect fluid or an anisotropic fluid is the usage of a geometric condition, known as the Karmarkar condition, which guarantees that the geometry (2 ) can be embedded in a 5 dimension pseudo euclidean manifold. The condition which guarantees that a four-dimensional curved space-time can be embedded in a five-dimensional pseudo-Euclidean space, which in the case of the metric given by the equation (2 ), is described in terms of the components from the Riemann's tensor by [70]8 ), and then from the algebra, we reach that the content of this condition is described by the differential equation:10 ) or (11 ), we would have the geometry determined. Since the integration of Einstein's equations does not guarantee that it represents a physically acceptable solution. One analysis of the solutions presented in the past century for the case of a perfect fluid shows that approximately 77 % of these are not physically acceptable [72], as such the proposal of physically acceptable solutions is not an easy task.
$\begin{eqnarray}{R}_{{trtr}}\,{R}_{\theta \phi \theta \phi }-{R}_{t\theta t\theta }\,{R}_{r\phi r\phi }=0,\end{eqnarray}$
with the restriction Rθφθφ ≠ 0 [71]. The Riemann components which appear in this relation are: $\begin{eqnarray*}\begin{array}{rcl}{R}_{{trtr}} & = & -\displaystyle \frac{{{\rm{e}}}^{2\nu }}{2}\left[2\,{\nu }^{{\prime\prime} }+2{\nu }^{{\prime} \ 2}-{\lambda }^{{\prime} }\,{\nu }^{{\prime} }\right],\\ {R}_{t\theta t\theta } & = & {R}_{t\phi t\phi }\,{\sin }^{2}\theta =-r\,{\nu }^{{\prime} }\,{{\rm{e}}}^{2\nu -\lambda },\\ {R}_{\theta \phi \theta \phi } & = & {{\rm{e}}}^{-\lambda }\,{r}^{2}\left(1-{{\rm{e}}}^{-\lambda }\right)\,{\sin }^{2}\theta ,\\ {R}_{r\phi r\phi } & = & {R}_{r\theta r\theta }\,{\sin }^{2}\theta =\displaystyle \frac{r}{2}\,{\lambda }^{{\prime} }.\end{array}\end{eqnarray*}$
Replacing these components from Rαβγδ in the equation ( $\begin{eqnarray}2\displaystyle \frac{\nu ^{\prime\prime} }{\nu ^{\prime} }+2\nu ^{\prime} -\displaystyle \frac{\lambda ^{\prime} }{1-{{\rm{e}}}^{-\lambda }}=0,\end{eqnarray}$
which relates the metric coefficients gtt = − e2λ(r) and grr = eν(r). That can be solved by assuming one of the functions λ or ν that are known, resulting in $\begin{eqnarray}{{\rm{e}}}^{\nu \left(r\right)}={C}_{1}+{C}_{2}\int \sqrt{{{\rm{e}}}^{\lambda \left(r\right)}-1}\,{\rm{d}}r,\end{eqnarray}$
$\begin{eqnarray}{{\rm{e}}}^{\lambda \left(r\right)}=1-C{{\rm{e}}}^{2\,\nu \left(r\right)}\nu {{\prime} }^{2},\end{eqnarray}$
the constants C, C1 and C2 being integration constants. So, proposing one of the metric functions, and if it is possible to integrate the respective equation (3. The solution
In the mechanism of the construction of solutions, through an imposition of the Karmarkar equation, it is very important to choose the adequate starting function, not only to obtain the integration of the system, but also to guarantee that it is physically acceptable. In our case we choose the function λ(r) given by12 ) is useful for generating new physically acceptable models. On the other hand, part of the reason for the functionality of this choice to describe the interior of the stars is the characteristics that satisfy λ, or grr = eλ, in a region r ∈ [0, a), near to the center of the star
$\begin{eqnarray}\lambda (r)=\mathrm{ln}\left[1+{{Car}}^{2}{\arctan }^{2}\left(d+{{ar}}^{2}\right)\right],\end{eqnarray}$
that was previously proposed in the description of solutions with anisotropic fluid [73] and for a perfect charged fluid [74]. Later on, it was employed in the representation of a model composed of ordinary matter and quintessence type matter [75], and these results increase the possibility that the function ( $\begin{eqnarray}{g}_{{\text{}}{rr}}(r)=1+{Ca}({\arctan }^{2}d){r}^{2}+\displaystyle \frac{2{{Ca}}^{2}(\arctan d){r}^{4}}{1+{d}^{2}}+{ \mathcal O }({r}^{6}),\end{eqnarray}$
and also grr(0) = 1 what prevents the existence of conical singularities in r = 0.One notorious difference with the previously analysed cases is that in these a state equation is not assumed, meanwhile in the one proposed here the model is constructed in a manner that the radial pressure and the density are related through the MIT Bag state equation. Substituting in the equation (10 ) the form of the function λ given by the equation (12 ) and integrating, we obtain the function:3 )–(5 ) the functions (λ, ν) given by the equations (12 ) and (14 ), and considering the state equation (15 ), after the algebra we obtain the functions:12 ) and (14 ) and the exterior geometry described by the Reissner Nordstrom metric:24 ) and the relation for Bg, equation (23 ), it's required that [20, 72]: (a) the behaviour of the density and pressure functions are positive, regular and monotonically decreasing functions; (b) that the causality is not violated and that the geometry is regular and absent of event horizon; (c) that the magnitude of the electric field is a monotonically increasing function and that it is zero on the center; (d) that the energy conditions are satisfied; (e) and that the solution is stable. These properties, for the solution, were shown considering the star Her X-1.
$\begin{eqnarray}\begin{array}{l}\nu (r)=\mathrm{ln}\left\{{\text{}}{C}_{1}+{\text{}}{C}_{2}\left[2(d+{{ar}}^{2})\arctan (d+{{ar}}^{2})\right.\right.\\ \,\left.\left.-\mathrm{ln}\left(1+{\left({{ar}}^{2}+d\right)}^{2}\right)\right]\right\}.\end{array}\end{eqnarray}$
This allowed us to determine the form of the geometry. Once known the metric functions we impose the MIT Bag state equation [54] $\begin{eqnarray}{P}_{r}(\rho )=\displaystyle \frac{1}{3}({c}^{2}\rho -4{B}_{g}),\end{eqnarray}$
to obtain the density, radial pressure, tangential pressure and electric field functions we substitute in ( $\begin{eqnarray}\begin{array}{rcl}{{kc}}^{2}\rho (r) & = & \displaystyle \frac{3\left[\left(1+{\left(d+{{ar}}^{2}\right)}^{2}\right)\arctan \left(d+{{ar}}^{2}\right)+2\,{{ar}}^{2}\right]{Ca}\arctan \left(d+{{ar}}^{2}\right)}{2{\left(1+{{Car}}^{2}{\left(\arctan \left(d+{{ar}}^{2}\right)\right)}^{2}\right)}^{2}\left({\left(d+{{ar}}^{2}\right)}^{2}+1\right)}\\ & & +\displaystyle \frac{6{\text{}}{C}_{2}\,a\arctan \left(d+{{ar}}^{2}\right){{\rm{e}}}^{-\nu }}{\left(1+{{Car}}^{2}{\left(\arctan \left(d+{{ar}}^{2}\right)\right)}^{2}\right)}+{{kB}}_{g},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}k{\text{}}{P}_{t}(r) & = & \displaystyle \frac{[8{C}_{2}{{\rm{e}}}^{-\nu }-3C\arctan (d+{{ar}}^{2})]\left[[1+{\left(d+{{ar}}^{2}\right)}^{2}]\arctan (d+{{ar}}^{2})+2{{ar}}^{2}\right]a}{2\left({\left(d+{{ar}}^{2}\right)}^{2}+1\right){\left(1+{{Car}}^{2}{\left(\arctan \left(d+{{ar}}^{2}\right)\right)}^{2}\right)}^{2}}\\ & & +\displaystyle \frac{a\arctan \left(d+{{ar}}^{2}\right)\left(10\,{C}_{2}{{\rm{e}}}^{-\nu }-\arctan \left(d+{{ar}}^{2}\right)C\right)}{1+{{Car}}^{2}{\left(\arctan \left(d+{{ar}}^{2}\right)\right)}^{2}}+{{kB}}_{g},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}E{\left(r\right)}^{2} & = & \displaystyle \frac{3\left[\left(1+{\left(d+{{ar}}^{2}\right)}^{2}\right)\arctan \left(d+{{ar}}^{2}\right)+2\,{{ar}}^{2}\right]{Ca}\arctan \left(d+{{ar}}^{2}\right)}{2{\left(1+{{Car}}^{2}{\left(\arctan \left(d+{{ar}}^{2}\right)\right)}^{2}\right)}^{2}\left({\left(d+{{ar}}^{2}\right)}^{2}+1\right)}\\ & & -\displaystyle \frac{\arctan \left(d+{{ar}}^{2}\right)a\left(6\,{\text{}}{C}_{2}\,{{\rm{e}}}^{-\nu }-\arctan \left(d+{{ar}}^{2}\right)C\right)}{1+{{Car}}^{2}{\left(\arctan \left(d+{{ar}}^{2}\right)\right)}^{2}}-{{kB}}_{g}.\end{array}\end{eqnarray}$
Now we must impose some of the physical requirements to be able to determine the form of the constants C, C1 and C2, these are basically properties in the union of the interior and exterior regions of the star r = R [76], specifically, the interior geometry determined by the functions ( $\begin{eqnarray}\begin{array}{l}{\rm{d}}{s}^{2}=-\left[1-\displaystyle \frac{2{GM}}{{c}^{2}r}+\displaystyle \frac{{Q}^{2}}{{r}^{2}}\right]{\rm{d}}{t}^{2}\\ \,+{\left[1-\displaystyle \frac{2{GM}}{{c}^{2}r}+\displaystyle \frac{{Q}^{2}}{{r}^{2}}\right]}^{-1}{\rm{d}}{r}^{2}+{r}^{2}{\rm{d}}{{\rm{\Omega }}}^{2},\end{array}\end{eqnarray}$
which must be continuous in the surface of the star, just like the second fundamental form. In addition to this, the electric charge in the center of the star and the anisotropy factor Δ(0) = Pt(0) − Pr(0) must be zero and the enclosed net charge must match the value of the charge Q in the Reissner Nordstrom metric. When we apply these requirements we obtain the constants: $\begin{eqnarray}C=\displaystyle \frac{2u-{q}^{2}}{\left(1-2u+{q}^{2}\right)\ {{aR}}^{2}\ {\arctan }^{2}\left(d+{{aR}}^{2}\right)},\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{\text{}}{C}_{1}=\displaystyle \frac{\left({q}^{2}-u\right)}{\sqrt{1-2u+{q}^{2}}}\left[\displaystyle \frac{\left(d+{{aR}}^{2}\right)}{2{{aR}}^{2}}\right.\\ \quad \left.-\displaystyle \frac{\mathrm{ln}\left[1+{\left(d+{{aR}}^{2}\right)}^{2}\right]}{4{{aR}}^{2}\arctan \left(d+{{aR}}^{2}\right)}\right]+\sqrt{1-2u+{q}^{2}},\end{array}\end{eqnarray}$
$\begin{eqnarray}{\text{}}{C}_{2}=-\displaystyle \frac{{q}^{2}-u}{4\sqrt{1-2u+{q}^{2}}\arctan \left(d+{{aR}}^{2}\right){{aR}}^{2}},\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{B}_{g}=-\displaystyle \frac{{q}^{2}\left({q}^{2}-4u+2\right)+u\left(4u-3\right)}{2{{kR}}^{2}}\\ \quad +\displaystyle \frac{a\left(2u-{q}^{2}\right)\left(1-2u+{q}^{2}\right)}{\left[1+{\left(d+{{aR}}^{2}\right)}^{2}\right]k\arctan \left(d+{{aR}}^{2}\right)},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}\left[1+{\left(d+{{aR}}^{2}\right)}^{2}\right]\left(2{{kR}}^{2}{B}_{g}+u(4u-3)\right)\\ \quad \times \arctan \left(d+{{aR}}^{2}\right)-4u\left(1-2u\right){{aR}}^{2}\\ \quad +\left[2\,[1+{\left(d+{{aR}}^{2}\right)}^{2}]\left(1-2u\right)\right.\\ \quad \times \left.\arctan \left(d+{{aR}}^{2}\right)+2\left(1-4u\right){{aR}}^{2}\right]{q}^{2}\\ \quad +\left[[1+{\left(d+{{aR}}^{2}\right)}^{2}]\arctan \left(d+{{aR}}^{2}\right)+2\,{{aR}}^{2}\right]{q}^{4}=0,\end{array}\end{eqnarray}$
where u = GM/c2R and q = Q/R. This last relation is a restriction between the dimensionless parameters (u, q, d, aR2). In addition to this restriction, which guided us to the determination of the constants (C, C1, C2), the restriction (4. Correspondence of the model with the observational data of the star Her X-1
Although in the analysis of the solution we can consider different values for the compactness u = GM/c2R, with the purpose of applying the solution to a case associated with observational data we will focus on the graphic analysis considering the observational data of the star Her X-1. The interval of observational values of the mass and radius are M = 0.85 ± 0.15M⊙ and R = 8.1 ± 0.41 km respectively, the greater compactness is obtained by contemplating the greater mass in the lower radius that for the star Her X-1 are M = 1M⊙ and R = 7.69 km which would correspond to a compactness u = 0.1920, for this value we will center our graphic analysis. Equation (24 ) shows that the values of the parameters and physical values (u, q, R, Bg, d, aR2) are not independent, however, for a specific value of u there are intervals of possible values of the Bag constant Bg and charge that satisfy the equation (24 ). Inside these intervals, we choose values of the Bag constant (charge Q) between the minimum and maximum for the graphic analysis. The graphs are presented in terms of dimensionless functions kc2R2ρ(r), kR2Pr(r), kR2Pt(r), ${R}^{2}E{\left(r\right)}^{2}$, ${v}_{t}{\left(r\right)}^{2}/{c}^{2}$ and γ(r). In figure 1, the positive and monotonically decreasing behaviour for the values of charge and Bag constant is shown. In it, we observe that for a greater value of Bg the density is greater and as Bg decreases the density also decreases. This situation was expected on the surface of the star (r = R) given the state equation ${P}_{r}(\rho )=\tfrac{1}{3}({c}^{2}\rho -4{B}_{g})$ and in this model, a behaviour similar continues to occur in the interior (as it can be observed from the fact that the graphs of the density do not overlap).
Figure 1. Behaviour of the density for different values of the Bag constant. |
The effect of the presence of the total accumulated charge in the interior of the star can be understood by starting from figures 2 and 3 for the radial and tangential pressures respectively, and in these we observe that for lower charge values the pressure is greater (purple line) and that when the charge is greater the pressure is lower, that is to say, the presence of a greater charge makes it so that the pressure diminishes counteracting the attractive gravitational effect. From these graphs, we also observe that both pressures (radial and tangential) are equal in the center of the star and that the radial pressure is zero on the surface of the star. The radial speed of sound is constant, given the MIT Bag equation of state for the radial pressure:
$\begin{eqnarray*}\displaystyle \frac{{v}_{r}^{2}}{{c}^{2}}=\displaystyle \frac{1}{{c}^{2}}\displaystyle \frac{\partial P(\rho )}{\partial \rho }=\displaystyle \frac{1}{3}.\end{eqnarray*}$
Meanwhile, the tangential speed of sound for the different values of charge and of MIT Bag constant is represented in figure 4 and it is observed that the causality condition is not violated. Another of the relevant functions is the electric field's magnitude function $E{\left(r\right)}^{2}$ which is represented in figure 5, and from it we observe that, as it's physically expected, the intensity of the electric field is zero at the center and that from greater values of the total charge Q the intensity of the electric field is greater.
Figure 2. Graphic representation of the radial pressure for different values of the electric charge. |
Figure 3. Graphic representation of the tangential pressure for different values of electric charge. |
Figure 4. The tangential speed of sound. |
Figure 5. Magnitude of the electric field for different values of the charge and Bag constant. |
4.1. Energy conditions and stability conditions
Other requirements that are imposed on the interior solutions to determine if these are physically acceptable are those called energy conditions. In the case of a charged anisotropic fluid, these are:
$\begin{eqnarray*}\begin{array}{l}{\rm{NEC}}:{c}^{2}\rho +{P}_{r}\geqslant 0,\\ \quad {{kc}}^{2}\rho +{{kP}}_{t}+2{E}^{2}\geqslant 0.\\ {\rm{WEC}}:{c}^{2}\rho +{P}_{r}\geqslant 0,\quad {{kc}}^{2}\rho +{E}^{2}\geqslant 0,\\ \quad {{kc}}^{2}\rho +{{kP}}_{t}+2{E}^{2}\geqslant 0.\\ {\rm{SEC}}:{c}^{2}\rho +{P}_{r}\geqslant 0,\quad {{kc}}^{2}\rho +{{kP}}_{r}+2{{kP}}_{t}+2{E}^{2}\geqslant 0,\\ \quad {{kc}}^{2}\rho +{E}^{2}\geqslant 0,\\ \quad {{kc}}^{2}\rho +{{kP}}_{t}+2{E}^{2}\geqslant 0.\\ {\rm{DEC}}:{c}^{2}\rho +{P}_{r}\geqslant 0,\quad {{kc}}^{2}\rho +{E}^{2}\geqslant 0,\\ \quad {c}^{2}\rho -{P}_{t}\geqslant 0,\quad {{kc}}^{2}\rho -{{kP}}_{r}+2{E}^{2}\geqslant 0,\\ \quad {{kc}}^{2}\rho +{{kP}}_{t}+2{E}^{2}\geqslant 0.\\ {\rm{TEC}}:{c}^{2}\rho -{P}_{r}-2{P}_{t}\geqslant 0.\end{array}\end{eqnarray*}$
Of course that, for the verification of these, it is not necessary to realize graphs for each one of these inequalities, it is enough to note that the density, pressures and magnitude of the electric field are positive to guarantee that the NEC, WEC, SEC and some of the inequalities of the DEC are satisfied. Also, we note that the density is greater than any of the pressures (radial or tangential) so the DEC is also satisfied. Here we only show in figure 6 the graph of the TEC. For the stability analysis, we will apply two criteria, the first of these, the Harrison–Zeldovich–Novikov criteria [77, 78], does indicate that if $\tfrac{\partial M}{\partial {\rho }_{c}}\gt 0$ (with ρc the central density) then we have guaranteed the stability of the gaseous stellar configuration in relation to radial pulsations. The mass M as a function of the central density given by $\begin{eqnarray}M\left({\rho }_{c},Q\right)=\displaystyle \frac{{c}^{2}\left({R}^{2}\left({Q}^{2}+{R}^{2}\right)\left({c}^{2}{\rho }_{c}-2{B}_{g}\right)k+3{Q}^{2}\right)}{{RG}\left(3+2\,{R}^{2}k\left({c}^{2}{\rho }_{c}-2\,{B}_{g}\right)\right)},\end{eqnarray}$
and calculating the derivative in relation to the central density: $\begin{eqnarray}\displaystyle \frac{\partial }{\partial {\rho }_{c}}M\left({\rho }_{c},Q\right)=\displaystyle \frac{3{{kc}}^{4}R\left({R}^{2}-{Q}^{2}\right)}{G{\left(3+2{{kR}}^{2}\left({c}^{2}{\rho }_{c}-2\,{B}_{g}\right)\right)}^{2}}\gt 0,\end{eqnarray}$
the positive sign is placed since R2 − Q2 > 0. The other criteria that we verify, as to determine whether it's satisfied or not, is the one of the adiabatic index, which in the case of anisotropic pressures, has the stability guaranteed in regards to infinitesimal radial adiabatic perturbation, if [79–83]: $\begin{eqnarray}\gamma =\displaystyle \frac{{c}^{2}\rho +{P}_{r}}{{P}_{r}}\displaystyle \frac{\partial {P}_{r}(\rho )}{\partial \rho }\gt {\gamma }_{{\rm{crit}}}=\displaystyle \frac{4}{3}+\displaystyle \frac{19}{21}u.\end{eqnarray}$
In this case, for the star Her X-1, u = 0.1920, as such the required condition is γ ≤ γcrit ≈ 1.507. Considering this value, in figure 7 it is shown that for each of the charge values and Bag constant, the required condition is satisfied, and as such the solution is stable according to the adiabatic index criteria. The continuity of the geometry is shown in figure 8. Here we note that the metric functions are continuous, including the surface of the star located in r = 7.69. We can also observe that the difference, for the different values of the electric charge, between the metric components is more noticeable for $-{g}_{{tt}}^{(i)}$ and that ${g}_{{rr}}^{(i)}(0)=1$ is a required condition for the regularity of the geometry.
Figure 6. Behaviour of the TEC for the different values of the Bag constant and electric charge with the compactness value u = 0.1920. |
Figure 7. Graphic representation of the adiabatic index for the star Her X-1, considering the compactness u = 0.1920 for which the critical adiabatic index is γcrit ≈ 1.507. |
Figure 8. Continuity of the metric coefficients that determine the interior geometry ($-{g}_{{tt}}^{(i)}$, $1/{g}_{{rr}}^{(i)}$ ) and the exterior geometry ($-{g}_{{tt}}^{(e)}=1/{g}_{{rr}}^{(i)}$ ). |
4.2. Hydrostatc equilibrium condition
The conservation of the energy momentum tensor ∇α Tαβ = 0 generates the relation that describes the hydrostatic equilibrium
$\begin{eqnarray}\begin{array}{l}-{\text{}}{P}_{r}^{\prime} -\left({\text{}}{P}_{r}+{c}^{2}\rho \right)\nu ^{\prime} +\displaystyle \frac{2}{r}({P}_{t}-{P}_{r})\\ \quad +2\displaystyle \frac{E}{{r}^{2}}[{r}^{2}E]^{\prime} ={F}_{h}+{F}_{g}+{F}_{{afe}}=0,\end{array}\end{eqnarray}$
where Fh, Fg and Fafe represent the force of the pressure gradient, the gravitational force and the effective anisotropic force (which is generated by the anisotropy in the pressures and the electric force) respectively, and these are given by $\begin{eqnarray}\begin{array}{rcl}{F}_{h} & = & -{P}_{r}^{{\prime} },\quad {F}_{g}=-\displaystyle \frac{{\nu }^{{\prime} }}{2}({c}^{2}\rho +P),\\ {F}_{{afe}} & = & \displaystyle \frac{2}{r}({P}_{t}-{P}_{r})+2\displaystyle \frac{E}{{r}^{2}}[{r}^{2}E]^{\prime} .\end{array}\end{eqnarray}$
The graphic representation of the forces is shown in figure 9. In this figure, we appreciate the attractive effect generated by the gravitational and repulsive force due to the pressure gradient, as well as the effective anisotropic force, which is one order of magnitude lower than the hydrostatic force.
Figure 9. Graphic behaviour of the forces present in the interior of the stars. |
5. Discussion and conclusions
In this work, we presented a model for compact stars constructed from the Karmarkar condition, taking as a starting point the supposition of the metric function ${g}_{{rr}}\,={\left[1+{{Car}}^{2}{\arctan }^{2}(d+{{ar}}^{2})\right]}^{-1}$. The choice of this function and the metric component gtt satisfying the Karmarkar condition implies an embedding of the geometry that satisfies Einstein's equations in a flat 5 dimension space. The fundamental reason why so many interior solutions constructed through this mechanism transpire to be physically acceptable is still an unsolved problem [55–67]. The substantial difference with other works that employ the previously mentioned approach lies in obtaining the relation between the Bag constant, the compactness of the stars and the electric charge, equation (23 ), as well as, just like other works, achieving results that are consistent with the physical characteristics and orders of magnitude from the physical quantities. Specifically, by means of the graphic analysis, it has been shown that for the maximum compactness of the star Her X-1 obtained from the observational data, the density, pressures and tangential speed of sound are positive, regular and monotonically decreasing functions, while the magnitude of the electric field is positive. The solution satisfies the energy conditions and is stable according to the Harrison–Zeldovich–Novikov stability criteria and the adiabatic index criteria. Also, it was shown that as a result of the presence of the charge, the pressure is lower when the net charge is greater, that is to say, the charge counteracts the attractive effect of the gravity. In table 1, we report the values of the central density ρc, the density on the surface of the star ρb, the central and tangential pressure on the surface for the values of the Bag constant and of the total charge with which the graphs were generated. The table shows that as the minimum and maximum Bag constant differs by around 4MeV/fm−3, the maximum charge is approximately 23 times greater than the minimum charge and that for a greater value of the Bag constant, the charge is lower. From the table, we also observe that the orders of magnitude of the Bag constant, electric charge, density and pressures are consistent with the typical orders of magnitude for this type of star.
Table 1. Values of the central density ρc, on the surface ρb pressure, speed of sound and for the different values of the Bag constant and net charge for the star Her X-1, for the values of radius R = 7.69 km and M = 1M⊙. |
| ${B}_{g}\left(\tfrac{\mathrm{MeV}}{{\mathrm{fm}}^{3}}\right)$ | $Q\left({10}^{18}{\rm{C}}\ \right)$ | ${\rho }_{c}\left({10}^{18}\tfrac{\mathrm{kg}}{{{\rm{m}}}^{3}}\right)$ | ${\rho }_{b}\left({10}^{18}\tfrac{\mathrm{kg}}{{{\rm{m}}}^{3}}\right)$ | ${P}_{{\text{}}{rc}}\left({10}^{34}\mathrm{Pa}\right)$ | ${P}_{{\text{}}{tb}}\left({10}^{34}\mathrm{Pa}\right)$ |
|---|---|---|---|---|---|
| 122.13 | 5.57 | 1.3077 | 0.87091 | 1.3085 | 0.3891 |
| 120.67 | 84.0 | 1.2331 | 0.86048 | 1.1160 | 0.2044 |
| 119.26 | 110.0 | 1.1806 | 0.85042 | 0.9893 | 0.0777 |
| 118.70 | 131.0 | 1.1353 | 0.84647 | 0.8651 | 0.0001 |
In this work, we addressed a model which considers that the interior is electrically charged and that in the interior the radial pressure satisfies the MIT Bag state equation adequate for the description of the stars that contain quarks in their interior. However, independently of the state equation considered, the geometry is determined starting from the Karmarkar condition. As such a question that might be addressed in future works is the analysis of some models for which, in functional terms, the metric is determined by the same Karmarkar geometric condition for the same metric potential grr but with another state equation and from it, do a comparative study that will allow us to discern which state equation is the most adequate for the description of a star's interior with the same functional form of the metric.