Compact stars in the non-minimally coupled electromagnetic fields to gravity
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Abstract
We investigate the gravitational models with the non-minimal \(Y(R)F^2\) coupled electromagnetic fields to gravity, in order to describe charged compact stars, where Y(R) denotes a function of the Ricci curvature scalar R and \(F^2\) denotes the Maxwell invariant term. We determine two parameter family of exact spherically symmetric static solutions and the corresponding non-minimal model without assuming any relation between energy density of matter and pressure. We give the mass-radius, electric charge-radius ratios and surface gravitational redshift which are obtained by the boundary conditions. We reach a wide range of possibilities for the parameters k and \(\alpha \) in these solutions. Lastly we show that the models can describe the compact stars even in the more simple case \(\alpha =3\).
1 Introduction
Spherically symmetric solutions in gravity are fundamental tools in order to describe the structure and physical properties of compact stars. There is a large number of interior exact spherically symmetric solutions of Einstein’s theory of gravitation (for reviews see [1, 2]). But, very few of them satisfy the necessary physical and continuity conditions for a compact fluid. Some of them were given by Mak and Harko [3, 4, 5] for an isotropic neutral spherically symmetric matter distribution.
A charged compact star may be more stable [6] and prevent the gravitationally collapse [7, 8], therefore it is interesting to consider the case with charge. The charged solutions of Einstein–Maxwell field equations which describe a strange quark star were found by Mak and Harko [9] considering a symmetry of conformal motions with MIT bag model.
Since the Einstein’s theory of gravity has significant observational problems at large cosmological scales [10, 11, 12, 13, 14, 15, 16, 17], one need to search new theories of gravitation which are acceptable even at these scales. Some compact star solutions in such modified theories as the hybrid metric-Palatini gravity [18] and Eddington-inspired Born–Infeld (EIBI) gravity [19], were studied numerically.
One can consider that a charged astrophysical object can described by the minimal coupling between the gravitational and electromagnetic fields known as Einstein–Maxwell theory. However, the above problems of Einstein’s gravity at large scales can also lead to investigate the \(Y(R)F^2\) type modification of Einstein–Maxwell theory [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30]. Such non-minimal modifications can also be found in [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42] to obtain more information on the interaction between electromagnetic and gravitational fields and all other energy forms. The non-minimal couplings also can arise in such compact objects as black holes, quark stars and neutron stars which have very high density gravitational and electromagnetic fields [42]. If the extreme situations are disappeared, that is far from the compact stars the model turns out to be the Einstein–Maxwell case.
Here we consider the non-minimal \(Y(R)F^2\) type modification to the Einstein–Maxwell theory and generalize the exact solutions for radiation fluid case \(k=1\) in [42] to the cases with \(k\ne 1\), inspired by the study [9]. We obtain the inner region solutions and construct the corresponding model which turns to the Einstein–Maxwell theory in the outer region. We note that the inner solutions recover the solution obtained by Misner and Zapolsky [43] for charge-less case and \(b=0\). We find the surface gravitational redshift, matter mass, total mass and charge in terms of boundary radius and the parameters k and \(\alpha \) via the continuity and boundary conditions.
We organize the paper as follows: In Sect. 2, we give the non-minimal \(Y(R)F^2\) gravity model and field equations in order to describe a compact fluid. In Sect. 3, we obtain exact static, spherically symmetric solutions of the model under the conformal symmetry and the corresponding Y(R) function. In Sect. 4, we determine the total mass, charge and gravitational redshift of the star in terms of boundary radius and the parameters of the model k and \(\alpha \). We summarize the results in the last section.
2 The model for a compact star
3 Spherically symmetric solutions under conformal symmetry
4 Continuity conditions
4.1 The simple model with \(\alpha =3\)
The dimensionless parameter k, the charge-radius ratio \(\frac{\kappa ^2Q^2}{r_b^2} \) and the surface redshift z obtained by using the observational mass M and the radius \(r_b\) for some neutron stars
Star | \(\frac{M}{r_b} \ (\frac{M_{\odot }}{km})\) | k | \(\frac{\kappa ^2Q^2}{r_b^2}\) | z (redshift) |
---|---|---|---|---|
EXO 1745-248 | \(\frac{1.4}{11} \) [57] | 1.047 | 0.082 | 0.098 |
4U 1820-30 | \(\frac{1.58}{9.11}\) [58] | 1.084 | 0.095 | 0.156 |
4U1608-52 | \(\frac{1.74}{9.3}\) [59] | 1.098 | 0.099 | 0.174 |
SAX J1748.9-2021 | \(\frac{1.78}{8.18}\) [60] | 1.135 | 0.110 | 0.217 |
5 Conclusion
We have extended the solutions of the previous the non-minimally coupled \(Y(R)F^2\) theory [42] to the case with \(\rho \ne 3p\) under the symmetry of conformal motions. In the case without any assumption of the equation of state, we have acquired one more parameter k in the solutions and the corresponding model. The pressure and energy density in the solutions are decreasing function with r in the interior of the star.
By matching the interior solution with the exterior Reissner–Nordström solution and applying the zero pressure condition at the boundary radius of star \(r=r_b\), we determine some physical properties of the star such as the ratio of the total mass and charge to boundary radius \(r_b\) and gravitational redshift depending on the parameters k and \(\alpha \).
We note that the total mass and charge increase with increasing k values and we have not reach an upper bound for k in the extended non-minimal model. But the increasing k values give an upper bound for the gravitational redshift, \(z \approx 0.732\), which is smaller than the more general restriction found in [56] for compact charged objects. It is interesting to note that each \(\alpha \) and k value in this model (28) determines a different non-minimally coupled theory and each theory with the different parameters gives different mass-radius, charge-radius ratios and gravitational redshift configuration. We calculated k values and the corresponding other quantities of the compact stars with the simple case \(\alpha =3\) via the some observed mass-radius values in Table 1. In this case, we also obtained the approximate equation of state (55) by fitting the \(p-\rho \) curve of the model. By comparing the fitting equation of state with the MIT bag model for \(\alpha =3\), we found the gravitational mass \(M=0.65 M_\odot \) which is smaller than the mass obtained for conformal symmetric quark stars [9] in the Einstein–Maxwell theory. However the mass in our model increases as \(\alpha \) increases. The non-minimally coupled model has the arbitrary parameters \(\alpha \) and k which can be set in order to be consistent with compact star observations. Even for \(\alpha =3\), each k value can describe a charged compact star.
Footnotes
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