Noether Symmetry Approach to the Non-Minimally Coupled $Y(R)F^2$ Gravity

We use Noether symmetry approach to find spherically symmetric static solutions of the non-minimally coupled electromagnetic fields to gravity. We construct the point-like Lagrangian under the spherical symmetry assumption. Then we determine Noether symmetry and the corresponding conserved charge. We derive Euler-Lagrange equations from this point-like Lagrangian and show that these equations turn out to be the differential equations derived from the field equations of the model. Also we give two new exact asymptotically flat solutions to these equations and investigate some thermodynamic properties of these black holes.

account the non-minimal couplings between electromagnetic fields and gravity like Y (R)F 2 -type.
The feature of the non-minimal theory is to have a large class of solutions such as the spherically symmetric [2-6] and cosmological solutions [7][8][9]. It is interesting to note that such non-minimal couplings, which are first order in R, obtained from a five dimensional Lagrangian via dimensional reduction [10][11][12][13] and they investigated in various aspects [14][15][16][17] such as the charge conservation and the relationship between electrical charge and geometry. The general couplings in R n F 2 -form applied to the generation of primordial magnetic fields in the inflation stage [7,8,[18][19][20][21][22]. Therefore, it is possible to consider the more general couplings in Y (R)F 2 form and their solutions can give us more information about the relation between electric charge and space-time curvature. Especially in the presence of medium with very high density electromagnetic fields, these couplings may arise and their effects can be significant even far from the source.
The Noether symmetry approach is one of the effective techniques to find solutions of a Lagrangian without using field equations. This symmetry approach allows us to find conserved quantities of a model by using the symmetry of the Lagrangian which is invariant along a vector field. Then the vector field can be determined by this symmetry and each symmetry of the Lagrangian gives a conserved quantity. This symmetry approach has been applied to f (R) gravity successfully to find out solutions and select the corresponding f (R) function which is compatible with the Noether symmetry [23][24][25].
This study is organized as follows: In the second section, we find the first order point-like action of the model for the spherically symmetric static metric and electric field. After we apply the Noether symmetry approach to the action, we obtain the system of partial differential equations. By solving the system, we find the Noether charge and the corresponding vector field for the nonminimal Y (R)F 2 model. In the third section, after we obtain Euler-Lagrange equations we give two new solutions to the equations and investigate some thermodynamic properties of the black hole solutions. Finally, we summarize the results in the last section.

II. NOETHER SYMMETRY APPROACH FOR THE NON-MINIMAL MODEL
Let us start with the following action of the non-minimally coupled electromagnetic fields to gravity [2, 3,22] By taking the variation of the action, and obtaining the field equations, we can find the solutions [2-6] for the spherically symmetric static metric Here the corresponding Ricci curvature scalar is Alternatively, we can find the solutions also from Noether symmetry approach by taking the following action of the non-minimally coupled model with the Lagrange multiplier λ Here variation of the action with respect to λ gives us R =R andR is defined as to eliminate the second order derivatives in the action via integration by parts. The variation with respect to R gives If we substitute (5) and (6) in the action (4), we obtain the following Lagrangian We see that Y R (R)F ∧ * F term in the Lagrangian has higher order derivatives which complicates the Noether approach. But, fortunately we have the following equation from the trace of the field which corresponds to the conservation of the energy-momentum tensor [8] and eliminates the higher order derivatives in the Lagrangian. By taking the electromagnetic tensor F , which has only the electric potential φ(r), the Lagrangian of the model is obtained as In the Lagrangian, the second order derivatives can be eliminated by integration by parts and it turns out to be the following point-like Lagrangian By considering the the configuration space Q which has the generalized coordinates q i ≡ {A, B, φ, R} and its tangent space T Q ≡ {q i , q i }, we look for the symmetries of the Lagrangian. Noether's theorem states that if the Lie derivative of a Lagrangian vanishes along a vector field X then X is a symmetry of the action and each symmetry of the action corresponds to a conserved quantity or first integral such as Then we take the Lie derivative of the point-like Lagrangian in the configuration space to find the first integral Here the derivatives α i can be written by the chain rule and the Lie derivative of the point-like Lagrangian gives us the following system of partial differential equations ∂α 1 ∂φ A solution to the system for an arbitrary Y (R) function can be found as Here c 1 , c 2 are arbitrary constants. Then the X vector field can be found as (27) and the the constant of motion (14) becomes

III. EULER-LAGRANGE EQUATIONS
In order to determine the non-minimal function and the metric functions, we calculate the Euler- for the Lagrangian (11). Then we obtain the following differential equations for A, B, φ, R, respectively: From (30) and (32) we find where q is an integration constant and it corresponds to the electric charge of the source. We note that the condition (33) can be found by taking the derivative of equation (31) with respect to r as in [6]. Then we have only the following differential equation (31) to solve which is same with the differential equation obtained from the field equations of the model in [3][4][5][6] for B = r 2 . Thus we show that these two different methods give the same differential equation (35). The conserved charge (28) of the model turns out to be for the Noether symmetry. We see that the conserved quantity involves the gravitational coupling constant κ 2 and the electric charge of the system q. We also calculate the energy function from and find By substituting the Ricci scalar (3) in the energy function (38), we find that the function is equal to zero, since equation (38) is nothing more than equation (35). Furthermore, we can choose B = r 2 without loss of generality then (35) becomes

A. Some New Solutions
In order to obtain solutions of the differential equation (39), we can choose the non-minimal function Y (R) that determines the strength of the coupling and find the metric function A(r) as a first method. Alternatively, we can choose possible geometries which are asymptotically flat and involve correction terms to the known Reissner-Nordstrom solution as a second method. Then we can find the corresponding non-minimal function Y (R). Here we consider the second method and we take the following metric function with the Yukawa-like correction term which gives the solution Then the Ricci scalar becomes R = a e r r 2 for the metric function and by taking the inverse function r = 2W (x), we can re-express the non-minimal function (42) in terms of R as Secondly, we choose another metric function with the Yukawa-like correction term which is also asymptotically flat. Then we obtain the following solution φ(r) = q r − a(r 3 + 6r 2 + 18r + 24)e −r qr , Y (r) = 8q 2 /κ 2 4q 2 − a(r 4 + 4r 3 + 12r 2 + 24r + 24)e −r . (46) We calculate the Ricci scalar for the second metric as R = ae −r and the inverse function r = lnx with x = a R . Then the non-minimal function becomes B. Some Thermodynamic Properties of the Solutions for the above metric functions (40) and (44). We give the variation of the temperatures with the event horizon radius r h in Fig. 1 for this non-minimal model and the Reissner-Nordstrom case. By using the entropy of black hole S = πr 2 h , we calculate the heat capacity from for the above two metric functions and obtain Heat capacity gives us information about thermal stability intervals and phase transition points of a black hole. Heat capacity must be positive and finite for a stable black hole. The points where heat capacity is zero give us the Type-1 instability and the points where heat capacity diverges give us Type-2 instability points which correspond to the second order phase transition for a black hole.
We plot also the heat capacity versus the horizon radius r h for the Reissner-Nordstrom solution and the non-minimal model in Figure 2 with different ranges to see these points clearly. Numerically, we can find upper bounds for the non-minimal parameter a as a = 0.16 for A 1 solution and a = 0.026 for A 2 solution with q = 0.4, to have a stable black hole. Furthermore, the type-2 instability points decrease from 0.7 to 0, while a increases from 0 to the upper bounds, respectively. Moreover, these upper bounds of the parameter a can increase to higher values as the electric charge increases.
On the other hand, the first law of thermodynamics is given by where τ is the trace of energy-momentum tensor obtaining from the gravitational field equation and it can be related with the work density [31]. In the minimal Einstein-Maxwell case, this relation (56) is automatically satisfied, since the trace of Maxwell energy-momentum tensor is zero. In contrast to the Einstein-Maxwell theory, the non-minimally coupled Y (R)F 2 theory has a non-vanishing trace of energy-momentum tensor. By taking κ 2 = 8π, the trace is found

FIG. 2:
The heat capacity versus the event horizon radius with q = 0.4, the dotted curve corresponds to the Reissner-Nordstrom case and the solid curve to the non-minimal case.
In order to investigate whether these metric functions (40) and (44) satisfy the Smarr formula we calculate the correction term as for the the metric functions. Then we firstly consider the electric potential (41) at the event horizon Thus the mass obtained from the Smarr formula (56) turns out to be and it is equal to the mass obtained from A(r h ) = 0. The Smarr formula (56) is also satisfied for the second solution (45) similarly, and the mass is found

IV. CONCLUSION
In this study, we have considered Noether symmetry approach to find spherically symmetric, static solutions of the non-minimally coupled Y (R)F 2 theory. By considering the point-like Lagrangian of the Y (R)F 2 theory with spherically symmetry, we have found a vector field satisfy Noether symmetry and the corresponding conserved quantity for any Y (R) function. We have also derived Euler-Lagrange equations from the point-like Lagrangian. Then we have shown that these equations are same with the equations derived from the field equations of the non-minimal model.
We have also given two exact asymptotically flat solutions and the corresponding non-minimal model. Then we have investigated some thermodynamic properties of these solutions such as the Hawking temperature and the heat capacity to determine the thermal stability intervals of the solutions. Furthermore we have shown that the solutions satisfy the the modified Smarr formula for the models with non-zero energy-momentum tensor. [