Anisotropic Stars in the Non-minimal $Y(R)F^2$ Gravity

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.

Additionally, the presence of a constant electric charge on the surface of compact stars may increase the stability [18] and protect them from collapsing [19,20]. 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 [21]. Also, the upper and lower limits for the basic physical quantities such as mass-radius ratio, redshift were derived for charged compact stars [22] and for anisotropic stars [23]. A regular charged solution of the field equations which satisfy physical conditions was found in [24] and the constants of the solution were fixed in terms of mass, charge and radius [25]. Later the solutions were extended to the charged anisotropic fluids [26,27]. Also, anisotropic charged fluid spheres are studied in D-dimensions [28].
Recently, the observational problems at astrophysical scales such as dark energy and dark matter [29][30][31][32][33][34][35][36] have caused to search new modified theories of gravitation. Then, in the presence of electromagnetic fields, the Einstein-Maxwell theory can also be modified with the coupling in the form, Y (R)F 2 [37][38][39][40][41][42][43][44][45][46][47]. Furthermore, the charged compact stars can be described by the nonminimal couplings [48,49]. Therefore in this study, we investigate the anisotropic compact stars in the non-minimal Y (R)F 2 model and find a class of exact analytical solutions. Then we obtain the total mass, total charge and gravitational boundary redshift by the parameters of our model and the boundary radius of the star.

II. THE MODEL FOR ANISOTROPIC STARS
We will obtain the field equations of our model for the anisotropic stars by varying the action integral with respect to independent variables; the orthonormal co-frame 1-form e a , the Levi-Civita connection 1-form ω ab , and the electromagnetic potential 1-form A, where κ is the coupling constant, 0 is the permittivity of free space, ∧ and * denote the exterior product and the Hodge map of the exterior algebra, respectively, 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 + ω a b ∧ e b is the torsion 2-form, λ 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 ∧ e 1 ∧ e 2 ∧ e 3 .
In this study, we use the SI units differently from the previous papers [48,49]. We write F = where c is the light velocity, E i is electric field and B i is magnetic field under the assumption of the Levi-Civita symbol 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 = −cρ e * e 0 + J i * e i where ρ 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 where u = δ 0 a e a is a time-like 1-form, v = δ 1 a e a is a space-like 1-form, p r is the radial component of pressure orthogonal to the transversal pressure p t , and ρ is the mass density for the anisotropic matter in the star. In this case it must be κ = 8πG/c 4 for the correct Newton limit. The nonminimal coupling function Y (R) is dimensionless. Consequently every term in our Lagrangian has the dimension of (energy)(length). Finally we notice that the dimension of λ a must be energy because torsion has the dimension of length.
After substituting the connection varied equation into the co-frame varied one, we obtain the modified Einstein equation for our model where ι denotes the interior product of the exterior algebra satisfying the duality relation, ι b e a = δ a b , through the Kronecker delta, R a b := dω a b +ω a c ∧ω c b is the Riemann curvature 2-form, R a := ι b R ba is the Ricci curvature 1-form, R := ι a R a is the curvature scalar, Y R := dY /dR, e ab··· := e a ∧ e b ∧ · · · and F bc := ι c ι b F . As the left hand side of the equation (3) constitutes the Einstein tensor, the right one is called the total energy momentum tensor for which the law of energy-momentum conservation is valid. For details one may consult Ref. [48].
The variation of the action according to the electromagnetic potential A yields the modified Maxwell equation We have also noticed that the Maxwell 2-form is closed dF = 0 from the Poincaré lemma, since it is exact form F = dA. Thus, the two field equations (3) and (4) define our model and we will look for solutions to them under the condition which removes the potential instabilities from the higher order derivatives for the non-minimal theory. Here the non-zero 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 [48]. We also notice that the trace of the modified Einstein equation (3), obtained by multiplying with e a ∧, produces an explicit relation between the Ricci curvature scalar, energy density and pressures

III. STATIC SPHERICALLY SYMMETRIC ANISOTROPIC SOLUTIONS
We propose the following metric for the static spherically symmetric spacetime and the Maxwell 2-form for the static electric field parallel to radial coordinate in (1+3) dimensions where 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.
The integral of the electric current density 3-form J sourcing the electric field gives rise to the electric charge inside the volume V surrounded by the closed spherical surface ∂V with radius r Here we used the Stoke's theorem. The components of the modified Einstein equation (3) reads three coupled nonlinear differential equations where prime stands for derivative with respect to r. We obtain one more useful equation by taking the covariant exterior derivative of (3) In what follows we assume the linear equation of state in the star where ω is a constant in the interval 0 < ω ≤ 1. Finally we calculate the crucial constraint (5) where the Ricci curvature scalar is A. Exact solutions with conformal symmetry We will look for solutions to these differential equations (10)- (15) assuming that the metric (7) admits a one-parameter group of conformal motions, since inside of stars can be described by using this symmetry [21,[50][51][52]. The symmetry is obtained by taking Lie derivative of the metric tensor g ab with respect to the vector field ξ, L ξ g ab = φ 0 g(r) g ab , for the arbitrary metric function g(r) and the following metric function f (r) with arbitrary constants a and φ 0 . Here we consider the metric function g(r) inspired by [21], where b and α are arbitrary parameters. With these choices, the curvature scalar (16) is calculated as We notice that it must be α > 2 and b = 0 in order for that the curvature scalar is nonzero and regular at the origin. If b = 0, then the curvature scalar R becomes zero and this leads to constant Y (R) in which case the model reduces to the minimal Einstein-Maxwell theory.
Then the system of equations (10)-(15) has the following solutions for the metric functions (17), (18) and the anisotropic pressure p r = ωρc 2 where the composite function Y(R(r)) have obtained in terms of r as Y (r) and we have defined After obtaining r(R) from (19) we rewrite explicitly the non-minimal coupling function in terms of R Since the exterior vacuum region is described by Reissner-Nordström metric satisfying R = 0, the non-minimal coupling function becomes Y (R) = c 0 . Therefore we can fix c 0 = 1 without loss of generality. Then the corresponding model becomes admitting the interior metric ds 2 in = −a 2 r 2 c 2 dt 2 + 3 1 + br α dr 2 + r 2 dΩ 2 (29) and the the Maxwell 2-form in the interior of star, where q(r) is obtained from (9) as On the other hand at the exterior, the model admit the Reissner-Nordström metric (32) and the which represents the electric field in the radial direction, E =rQ/4π 0 r 2 , where Q is the total electric charge of the star which is obtained by writing r = r b in (31). We will be able to determine some of the parameters from the matching and the continuity conditions, and the others from the observational data.

B. Matching conditions
We will match the interior metric (29) and the exterior metric (32) at the boundary of the star r = r b for continuity of the gravitational potential, These equations are solved for a and b appeared in the interior metric functions Vanishing of the radial pressure at the boundary, p r (r b ) = ωc 2 ρ(r b ) = 0 in (20), determines the The behavior of the pressures and energy density is given in Fig. 1 in terms of the radial distance r for k = 1. Moreover the decreasing behavior of the quantities does not change for k = 1. We can 1 Here we use the SI units differently from [48,49] in which leads to F = Q r 2 e 1 ∧ e 0 and κQ 2 r 2 term in the metric. obtain an upper bound of the parameter k using the non-negative tangential pressure p t (r) in (22).
In order to obtain the bound we need to determine the interval of ω. Since the radial component of the sound velocity dpr dρ is non-negative and should not be bigger than the square of light velocity c 2 for the normal matter, the parameter ω takes values in the range 0 ≤ ω ≤ 1. We see from the which turns out to be for the case ω = 1, that leads to the minimum p t (r) function. Thus the parameter k must be k ≤ 1 for the non-negative tangential pressure. On the other hand, the total electric charge (43) must be a real valued exponential function, then we obtain the inequality which leads to k ≥ 2 α . Thus k must take values in this range and we find the maximum lower bound for the parameter k as k > 2 3 for α > 3 (which leads to positive gravitational redshift).
While the interior side of the star has the electrically charged matter distribution, the outer side is vacuity. Then the excitation 2-form G = Y F in the interior becomes the Maxwell 2-form G = F at the exterior. Therefore the interior electric charge q(r) which found from 0 cd * Y F = 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 where we eliminated the parameter b via (38). The total electric charge is shown as a function of α in Fig. 2 for ω = 0.1 (a) and ω = 0.3 (b) taking some different k values. We see that the increasing k values increase the total charge values. The substitution of (38) to (36) allows us writing the total mass of the star in terms of its total charge Then from (43) the total mass becomes We depict the graph of the total mass as a function of α in Fig. 3 for ω = 0.15 (a) and ω = 0.5 Additionally, the gravitational surface redshift defined by z : We see that the gravitational redshift is independent of the parameters ω and k then the limit α → ∞ gives the upper bound for the redshift, z < 0.732. On the other hand, α = 3 gives z = 0.
Then α must be α > 3 for the observational requirements. Variation of the surface redshift is shown in Fig. 4.

FIG. 4:
The gravitational surface redshift versus the parameter α.
C. The Simple Model with α = 4 The model simplifies for α = 4 as follows where X = 8kω + 12k − 10ω − 6 for α = 4. Here we emphasize that the non-minimal function in this model can be expanded Maclaurin series as for XR 2 144bω(k+1) < 1. Then the model admits the interior metric with the energy density, tangential pressure and electric field from (20), (22), (23) and (29) In this case the model gives only one redshift which is z = 0.225 from (46)  Now we focus on the case with k = 1 in which the equation of state must satisfy the special constraint, ρc 2 = p r + 2p t because of (6). Now we compute the associated quantities by using the equations (20), (22), (23) with the interior metric (29). The non-minimal coupling function (27) becomes explicitly .
We also calculate the total charge inside of the sphere with radius r eliminating b from (31) We check that the charge is regular at the origin for α > 2. Then we can find the the total charge and mass in terms of the boundary radius r b , the parameters ω and α from (43) and (45).
Variation of the total mass and electric charge as a function of the parameter α is shown in Fig.   5    for some neutron stars with ω = 0.2 and k = 1 .

IV. 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 = wρc 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 2 3 for the real valued total mass and electric charge. The total mass and electric charge increases with the increasing w values, while the gravitational redshift does not change. The total mass-boundary radius ratio has the upper bound GM c 2 r b = 0.48 which is grater than the Buchdahl bound [53] and the bounds given by [22,48]. The gravitational redshift at the surface only depends on the parameter α and increases with increasing α values up to the limit z = 0.732, which is the same result obtained from the isotropic case with k = 1 [48]. We also investigated some sub cases