# Anisotropic stars in the non-minimal \(Y(R)F^2\) gravity

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## Abstract

We investigate anisotropic compact stars in the non-minimal \(Y(R)F^2\) model of gravity which couples an arbitrary function of curvature scalar *Y*(*R*) to the electromagnetic field invariant \(F^2\). After we obtain exact anisotropic solutions to the field equations of the model, we apply the continuity conditions to the solutions at the boundary of the star. Then we find the mass, electric charge, and surface gravitational redshift by the parameters of the model and radius of the star.

## 1 Introduction

Compact stars are the best sources to test a theory of gravity under the extreme cases with strong fields. Although they are generally considered as isotropic, there are important reasons to take into account anisotropic compact stars which have different radial and tangential pressures. First of all, the anisotropic spherically symmetric compact stars can be more stable than the isotropic ones [1]. The core region of the compact stars with very high nuclear matter density becomes more realistic in the presence of anisotropic pressures [2, 3]. Moreover the phase transitions [4], pion condensations [5] and the type 3A superfluids [6] in the cooling neutron matter core can lead to anisotropic pressure distribution. Furthermore, the mixture of two perfect fluid can generate anisotropic fluid [7]. Anisotropy can be also sourced by the rotation of the star [8, 9, 10]. Additionally, strong magnetic fields may lead to anisotropic pressure components in the compact stars [11]. Some analytic solutions of anisotropic matter distribution were studied in Einsteinian Gravity [8, 9, 10, 11, 12, 13, 14, 15, 16, 17]. Recently it was shown that the “scalarization” can not arise without anisotropy and the anisotropy range can be determined by observations on binary pulsar in the Scalar-Tensor Gravity and General Relativity [10]. The anisotropic star solutions in \(R^2\) gravity can shift the mass-radius curves to the region given by observations [18]. It is interesting to note that anisotropic compact stars were investigated in Rastall theory and found exact solutions which permit the formation of super-massive star [19].

Additionally, the presence of a constant electric charge on the surface of compact stars may increase the stability [20] and protect them from collapsing [21, 22]. The charged fluids can be described by the minimally coupled Einstein–Maxwell field equations. An exact isotropic solution of the Einstein–Maxwell theory were found by Mak and Harko describing physical parameters of a quark star with the MIT bag equation of state under the existence of conformal motions [23]. Also, the upper and lower limits for the basic physical quantities such as mass-radius ratio, redshift were derived for charged compact stars [24] and for anisotropic stars [25]. A regular charged solution of the field equations which satisfy physical conditions was found in [26] and the constants of the solution were fixed in terms of mass, charge and radius [27]. Later the solutions were extended to the charged anisotropic fluids [28, 29]. Also, anisotropic charged fluid spheres were studied in *D*-dimensions [30].

In the investigation of compact stars, one of the most important problem is the mass discrepancy between the predictions of nuclear theories [31, 32, 33, 34] and neutron star observations. The resent observations such as \(1.97M_\odot \) of the neutron star PSR J1614-2230 [35], \(2.4M_\odot \) of the neutron stars B1957+20 [36] and 4U 1700-377 [37] or \(2.7M_\odot \) of J1748-2021B [38] can not be explained by using any soft equation of state in Einstein’s gravity [31, 35, 39]. Since each different approach which solves this problem leads to different maximal mass, we need to more reliable models which satisfy observations and give the correct maximum mass limit of compact stars.

On the other hand, the observational problems such as dark energy and dark matter [40, 41, 42, 43, 44, 45, 46, 47] at astrophysical scales have caused to search new modified theories of gravitation such as *f*(*R*) gravity [48, 49, 50, 51, 52, 53, 54]. As an alternative approach, the *f*(*R*) theories of gravitation can explain the inflation and cosmic acceleration without exotic fields and satisfy the cosmological observations [55, 56]. Therefore, in the strong gravity regimes such as inside the compact stars, *f*(*R*) gravity models can be considered to describe the more massive stars [57, 58] . Furthermore, the strong magnetic fields [59] and electric fields [60] can increase the mass of neutron stars in the framework of \(f(R)\) gravity.

On the other hand, in the presence of the strong electromagnetic fields, the Einstein–Maxwell theory can also be modified. The first modification is the minimal coupling between *f*(*R*) gravity and Maxwell field as the \(f(R)-\)Maxwell gravity which has only the spherically symmetric static solution [61, 62]. Therefore we consider the more general modifications such as \(Y(R)F_{ab}F^{ab}\) which allow a wide range of solutions [63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73].

A similar kind of such a modification which is \( R_{abcd}F^{ab} F^{cd} \) firstly was defined by Prasanna [74] and found a criterion for null electromagnetic fields in conformally flat space-times. Then the most general possible invariants which involves electromagnetic and gravitational fields (vector-tensor fields) including \(RF_{ab}F^{ab}\) term were studied in [75] and the spherically symmetric static solutions were obtained [76] for the unique composition of such couplings. Also such non-minimal modifications were derived by dimensional reduction of the five dimensional Gauss–Bonnet action [77, 78] and \(R^2\) gravity action [79, 80]. These invariant terms were also obtained by Feynman diagram method from vacuum polarization of the photon in the weak gravitational field limit [81]. The general form of the couplings such as \(R^nF^2\) were used to explain the seed magnetic fields in the inflation and the production of the primeval magnetic flux in the universe [67, 70, 71, 72, 73, 82].

Thus it is natural to consider the more general modifications which couple a function of the Ricci scalar with Maxwell invariant as \(Y(R)F^2\) form inside the strong electromagnetic and gravitational fields such as compact astrophysical objects. The more general modifications have static spherically symmetric solutions [64, 65, 66, 70] to describe the flatness of velocity curves of galaxies, cosmological solutions to describe accelerating expansion of the universe [63, 67, 68, 69] and regular black hole solutions [83]. Furthermore, the charged, isotropic stars and radiation fluid stars can be described by the non-minimal couplings [84, 85]. Therefore in this study, we investigate the anisotropic compact stars in the non-minimal \(Y(R)F^2\) model and find a family of exact analytical solutions. Then we obtain the total mass, total charge and gravitational surface redshift by the parameters of the model and the boundary radius of the star.

## 2 The model for anisotropic stars

*A*,

*R*is the curvature scalar,

*Y*(

*R*) is a function of

*R*representing the non-minimal coupling between gravity and electromagnetism,

*F*is the electromagnetic 2-form, \(F=dA\),

*J*is the electromagnetic current 3-form, \(L_{mat}\) is the matter Lagrangian 4-form, \(T^a := de^a + \omega ^a{}_b \wedge e^b\) is the torsion 2-form, \(\lambda _a\) Lagrange multiplier 2-form constraining torsion to zero and

*M*is the differentiable four-dimensional manifold whose orientation is set by the choice \(*1=e^0 \wedge e^1 \wedge e^2 \wedge e^3\).

*c*is the light velocity, \(E_i\) is electric field and \(B_i\) is magnetic field under the assumption of the Levi–Civita symbol \(\epsilon _{0123}=+1\). We adhere the following convention about the indices; \(a,b,c = 0,1,2,3\) and \(i,j, k = 1,2,3\). We also define \(J=-c \rho _e *e^0 + J_i *e^i\) where \(\rho _e\) is the charge density and \(J_i\) is the current density. We assume that the co-frame variation of the matter sector of the Lagrangian produces the following energy momentum 3-form

*Y*(

*R*) is dimensionless. Consequently every term in our Lagrangian has the dimension of (

*energy*)(

*length*). Finally we notice that the dimension of \(\lambda _a\) must be

*energy*because torsion has the dimension of

*length*.

*A*yields the modified Maxwell equation

*k*is a dimensionless constant. The case of \(k=0\) leads to \(Y(R)= constant\) corresponding to the minimal Einstein–Maxwell theory which will be considered as the exterior vacuum solution with \(R=0\). We will see from solutions of the model that the total mass and total charge of the anisotropic star are critically dependent on the parameter

*k*. The additional features of the constraint (5) may be found in [84]. We also notice that the trace of the modified Einstein equation (3), obtained by multiplying with \(e^a \wedge \), produces an explicit relation between the Ricci curvature scalar, energy density and pressures

## 3 Static spherically symmetric anisotropic solutions

*f*and

*g*are the metric functions and

*E*is the electric field in the radial direction which all three functions depend only on the radial coordinate

*r*.

*J*sourcing the electric field gives rise to the electric charge inside the volume

*V*surrounded by the closed spherical surface \(\partial V\) with radius

*r*

*r*. We obtain one more useful equation by taking the covariant exterior derivative of (3)

### 3.1 Exact solutions with conformal symmetry

*g*(

*r*) and the following metric function

*f*(

*r*)

*a*and \(\phi _0\). Here we consider the metric function

*g*(

*r*)

*b*and \(\alpha \) are arbitrary parameters. With these choices, the curvature scalar (16) is calculated as

*R*becomes zero and this leads to constant

*Y*(

*R*) in which case the model reduces to the minimal Einstein–Maxwell theory.

*Y*(

*R*(

*r*)) have obtained in terms of

*r*as

*Y*(

*r*) and we have defined

*r*(

*R*) from (19)

*q*(

*r*) is obtained from (9) as

^{1}

*Q*is the total electric charge of the star which is obtained by writing \(r=r_b\) in (30). We will be able to determine some of the parameters from the matching and the continuity conditions, and the others from the observational data.

### 3.2 Matching conditions

*a*and

*b*appeared in the interior metric functions

*b*

*r*for \(k=1\). Moreover the decreasing behavior of the quantities does not change for \(k\ne 1\). We can obtain an upper bound of the parameter

*k*using the non-negative tangential pressure \(p_t(r)\) in (21). In order to obtain the bound we need to determine the interval of \(\omega \). Since the radial component of the sound velocity \( \frac{d p_r}{d\rho } \) is non-negative and should not be bigger than the square of light velocity \(c^2\) for the normal matter, the parameter \(\omega \) takes values in the range \(0\le \omega \le 1\). We see from the Fig. 1b that the tangential pressure curves decrease for the increasing \(\omega \) values and the minimum curve can be obtained from the case with \(\omega = 1\). Then we obtain the following inequality from the first part of \(p_t(r)\)

*k*must be \(k \le 1\) for the non-negative tangential pressure. On the other hand, the total electric charge (42) must be a real valued exponential function, then we obtain the inequality

*k*must take values in this range

*k*as \(k>\frac{2}{3}\) for \(\alpha >3 \) (which leads to positive gravitational redshift).

*q*(

*r*) which found from \(\epsilon _0 c d*YF = J \) must be equal to the total electric charge

*Q*obtained from \( d* F = 0 \) at the exterior. Thus the continuity of the excitation 2-form at the boundary leads to \(q(r_b)= Q\) as the last matching condition

*b*via (37). The total electric charge is shown as a function of \(\alpha \) in Fig. 2 for \(\omega =0.1 \) (a) and \(\omega =0.3 \) (b) taking some different

*k*values. We see that the increasing

*k*values increase the total charge values. The substitution of (37)–(35) allows us writing the total mass of the star in terms of its total charge

*k*values.

We see that the gravitational redshift is independent of the parameters \(\omega \) and *k* then the limit \(\alpha \rightarrow \infty \) gives the upper bound for the redshift, \(z < 0.732\). On the other hand, \(\alpha =3\) gives \(z=0\). Then \(\alpha \) must be \(\alpha >3 \) for the observational requirements. Variation of the surface redshift is shown in Fig. 4.

*k*from (44). If we can also predict the total charge of star, we can fix \( \omega \) and

*k*separately from (42). For example, when we take the gravitational redshift of the neutron star EXO 0748-676 as \(z=0.35\) with \(M=2M_{\odot }\) and \(R=13.1\) km [90], we find that \(\alpha \approx 5.08 \) from (45) and there is one free parameter

*k*or \(\omega \) that must be fixed in (44). Then we can fix \(k=1\) which leads to \(\omega \approx 0.01\).

### 3.3 The Simple Model with \(\alpha =4 \)

*k*and \(\omega \) for each observational mass value

*M*and boundary radius \(r_b\) to describe the star.

### 3.4 The special case with \(k =1\)

*r*eliminating

*b*from (30)

*M*and boundary radius \(r_b\) ratio \(\frac{M}{r_b}\) to describe compact stars with the model. In future, the observations with different redshifts in Table 1 lead to different \(\omega \) and

*k*values with the corresponding \(\alpha \).

The dimensionless parameter \(\alpha \), the dimensionless charge-radius ratio \(\frac{\kappa ^2Q^2}{16\pi ^2 \epsilon _0 r_b^2} \) and the surface redshift *z* obtained by using the observational mass *M* and the radius \(r_b\) for some neutron stars with \(\omega =0.2 \) and \(k=1 \)

## 4 Conclusion

We have studied spherically symmetric anisotropic solutions of the non-minimally coupled \(Y(R)F^2\) theory. We have established the non-minimal model which admits the regular interior metric solutions satisfying conformal symmetry inside the star and Reissner–Nordstrom solution at the exterior assuming the linear equation of state between the radial pressure and energy density as \(p_r = \omega \rho c^2 \). We found that the pressures and energy density decrease with the radial distance *r* inside the star.

We matched the interior and exterior metric, and used the continuity conditions at the boundary of the star. Then we obtained such quantities as total mass, total charge and gravitational surface redshift in terms of the parameters of the model and the boundary radius of the star. We see that the parameter *k* can not be more than 1 for non-negative tangential pressure and can not be less than \(\frac{2}{\alpha } \) with \(\alpha >3 \) for the real valued total mass and electric charge. The total mass and electric charge increases with the increasing \(\omega \) values, while the gravitational redshift does not change. The total mass-boundary radius ratio has the upper bound \(\frac{GM}{c^2r_b} = 0.48\) which is grater than the Buchdahl bound [94] and the bounds given by [24, 84]. The gravitational redshift at the surface only depends on the parameter \(\alpha \) and increases with increasing \(\alpha \) values up to the limit \(z= 0.732\), which is the same result obtained from the isotropic case with \(k=1\) [84]. We also investigated some sub cases such as \(\alpha =4, k\ne 1\) and \(\alpha >3, k=1\), which can be model of anisotropic stars.

We note that an interesting investigation of anisotropic compact stars was recently given by Salako et al. [19] in the non-conservative theory of gravity. Then, the constants of the interior metric were determined for some known masses and radii. Then the physical parameters such as anisotropy, gravitational redshift, matter density and pressures were calculated and found that this model can describe even super-massive compact stars. On the other hand, in our conservative model (see [84] for energy-momentum conservation), \(\alpha \) can be determined by the gravitational surface redshift observations due to (45). Furthermore, if we can also determine the mass-boundary radius ratio \(\frac{M}{r_b}\) from observations, we can fix one of the two parameters \(\omega \) and *k* via (44). Additionally, the observation of the total charge can fix both of the parameters \(\omega \) and *k* from (42). Thus we can construct a non-minimal \(Y(R)F^2\) model for each charged compact star. But, the gravitational redshift measurements [89, 90] and total charge predictions do not have enough precise results. However we can predict possible ranges of these quantities.

## Footnotes

## Notes

### Acknowledgements

In this study the authors Ö.S. and F.Ç. were supported via the project number 2018FEBE001 and M.A. via the project number 2018HZDP036 by the Scientific Research Coordination Unit of Pamukkale University

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