Black-hole solution in nonlinear electrodynamics with the maximum allowable symmetries

The nonlinear Maxwell Lagrangian preserving both conformal and SO(2) duality-rotation invariance has been introduced very recently. Here, in the context of Einstein's theory of gravity minimally coupled with this nonlinear electrodynamics, we obtain a black hole solution which is the Reissner-Nordstr\"{o}m black hole with one additional parameter that is coming from the nonlinear theory. We employ the causality and unitarity principles to identify an upper bound for this free parameter. The effects of this parameter on the physical properties of the black hole solution are investigated.


I. INTRODUCTION
There are different models for nonlinear electrodynamics. The first such model, known as Born-Infeld (BI) nonlinear electrodynamics, which was fully relativistic and gauge invariant, proposed by Max Born and Leopold Infeld in 1934 [1]. The initial idea was to modify the Maxwell's linear Lagrangian i.e., L = − 1 4 F µν F µν to construct a nonlinear Lagrangian with respect to the Maxwell's invariants S = 1 2 F µν F µν and P = 1 2 F µνF µν such that the self-energy and the fields of a point charge remain finite at the location of the charge. Furthermore, the vacuum polarization phenomena in quantum electrodynamics (QED) has been observed experimentally since 1940s. It is the polarization of virtual electron-positron pairs in vacuum which is an indication for the nonlinear interaction of electromagnetic fields such as photon-photon scattering. The interaction between photons can be explained using the so-called Heisenburg-Euler (HE) effective-field theory. The HE model was proposed by W. Heisenburg and H. Euler in 1936 [2] and is valid in the weak-field limit and large wavelengths. There are other nonlinear electrodynamic models that have been introduced more recently. For instance, the Logarithmic [3], the Maxwell Power Law [4,5], the arcsin [6], the rational [7], the exponential [8] and the double-Logarithmic [9] models are among them which all reproduce Maxwell's linear model in the weak-field limit. Furthermore, there are NED models that don't reduce to the linear one in the weak-filed limit. Such models have been coupled to Einstein's theory for constructing regular electric black holes [10,11]. As it was proved by Bronikove [12], unlike the existence of a regular magnetic black hole, a regular electric black hole solution doesn't exist in the gravity coupled with a NED which yields Maxwell's theory in the weak-field limit.
In general, a generic NED model doesn't admit the symmetries of Maxwell's theory. Among them are preserving conformal and SO(2) duality-rotation invariance symmetries. In Ref. [13] a NED model has been introduced which respects these symmetries (see Eq. (1) below). In this interesting model, there is also a constant γ, which is, in accordance with [13] and [14], a positive parameter. In this study, we would like to apply the so-called causality and unitarity principles for making an estimation for the upper bound of the parameter γ. We would also like to examine the effects of this parameter in the physical properties of the black hole solution in the context of gravity coupled with this specific NED model.
Finally, it is worth to mention that, it is the conformal invariant symmetry of the Maxwell theory which results in a traceless energy-momentum tensor i.e., T µ µ = 0. The same symmetry in a NED field theory also yields a traceless energy-momentum tensor. This fact has been studied in [4] as well as in [15,16].
Our Letter is organized as follows. In Sec. II we present the NED model that admits conformal and SO(2) dualityrotation invariance symmetries. In Sec. III we find the black hole solution of the gravity minimally coupled with this NED. In Sec. IV we study the thermal stability of the solution. We conclude our work in Sec. V.

II. THE MODEL
The nonlinear Maxwell's Lagrangian is given by which has been first proposed in [13] and then re-proposed in [14]. Considering the electromagnetic two-form, given by in which is the electromagnetic field tensor and is the gauge potential one-form, the Maxwell invariants are defined to be S = 1 2 F µν F µν and P = 1 2 F µνF µν wherẽ is the Hodge dual two-form of F withF µν = 1 2 µναβ F αβ . In accordance with [13] and [14], γ is a positive parameter, however, we would like to see its possible upper bound by applying the causality and unitarity conditions. In accordance with the causality principle, the group velocity of the elementary electromagnetic excitations should be less than the speed of light in the vacuum and therefore there will be no tachyons in the theory spectrum. Also the unitarity principle requires the positive definiteness of the norm of every elementary excitation of the vacuum upon which ghosts are avoided. Following [17], these principles simply become, and Redefining L (S, P) = −Sy (z) with and z = P S , these inequalities reduce to y − zy ≥ 2y ≥ 0 for S < 0 and for S > 0. Considering the explicit form of y (z) we find and Clearly, with γ > 0, y − zy > 0 while y is definite negative which in turn imply that with S < 0 the causality and unitarity conditions are not satisfied. Hence, we must consider S > 0 upon which (10) must be held. In Fig. 1 we plot K = y − zy + 2z 2 y in terms of x for different values of γ. Our numerical calculation shows that for P > 0 and S > 0, (10) satisfied provided 0 < γ < 0.8814. Let's note that, γ is a dimensionless parameter of the theory which is bounded from above and below. In the rest of the paper, we shall consider γ to be in this interval.

FIG. 1:
Plot of K = y − zy + 2z 2 y in terms of z = P S for γ = 0.0000 to γ = 1.0000 with equal steps (= 0.1000) from top to bottom. K remains positive as long as γ 0.8814.

III. THE FIELD EQUATIONS AND THE BLACK HOLE SOLUTION
The action of Einstein-nonlinear-Maxwell theory is given by (G = 1) in which L (S, P) is given by Eq. (1). Upon applying the causality and unitarity conditions we have already obtained the upper limit for i.e., γ < 0.8814. Moreover, which is the linear Maxwell theory. The static spherically symmetric spacetime and the electromagnetic two-form are chosen to be and respectively, in which E and B are the radial components of the static electric and magnetic fields indicating the presence of the electric and magnetic monopols. Variation of the action with respect to the metric tensor implies the Einstein-nonlinear Maxwell equations given by in which is the energy-momentum tensor and G ν µ is the standard Einstein tensor. We note that, L S = ∂L ∂S and L P = ∂L ∂P . Furthermore, the variation of the action with respect to the four-potential yields the Maxell-nonlinear equations whereF is the dual two-form of F which is found to bẽ Having, F andF given by (16) and (20) we obtain and in which Q e is a constant satisfying which is a constant. The explicit form of the Maxwell's invariants are given by and Following the nonlinear-Maxwell equations, we shall solve the Einstein-nonlinear Maxwell equations. To do so, we find the nonzero componetnts of the energy momentum-tensor given by and Using a fluid model for the energy momentum tensor i.e., T ν µ = diag (−ρ, p r , p θ , p φ ) one finds Having, which is definite positive for all values of γ and x, we obtaine ρ ≥ 0 and ρ + p i ≥ 0 which in turn imply that the weak energy conditions are satisfied. Furthermore, the strong energy conditions i.e., ρ + p i ≥ 0 and ρ + i p i ≥ 0 are also satisfied. Next, we introduce in which definite positive. This is because of the causality and unitarity conditions upon which we imposed S > 0 or equivalently 0 < q = Qe Qm < 1. Hence, the energy momentum-tensor simplifies as Finally, the Einstein-nonlinear-Maxwell equations admits in which M is an integration constant, representing the mass of the black hole. This is a Reissner-Nordstrom-type [18] charged black hole solution with an additional parameter γ. In the next section, we study the effects of the parameter γ in the thermal stability of the black hole.

IV. THERMAL STABILITY OF THE BLACK HOLE SOLUTION
To investigate the effects of the parameter γ in the thermal stability of the black hole solution (39) we start with the Hawking temperature which is given by in which r h is the radius of the event horizon and Q 2 = Q 2 m + Q 2 e . In Fig. 2, we plot the 4πQ m T H versus x = r h Qm with Qe Qm = 0.2 and γ = 0.0000 to γ = 0.8814 with equal steps. Increasing the value of γ, for a given radius of the event horizon, increases the Hawking temperature. Furthermore, the heat capacity for constant Q is defined to be where S = πr 2 h is the entropy of the black hole. In Fig. 3 we plot C Q /Q 2 m with respect to x = r h Qm with Qe Qm = 0.2 and γ = 0.0000 to γ = 0.8814 with equal steps. The Type-1 (C Q = 0) and Type-2 (C Q → ±∞) transition points are emphasized. These points are given by and Let's add that the thermal stability region is defined to admit both T H and C Q positive. Therefore, the black hole is thermally stable if (r h ) T ype−1 < r h < (r h ) T ype−2 . It is also observed from the Fig. 3 that, the transition points are shifted to the smaller values for larger γ which in turn yields narrower stability region.

V. CONCLUSION
We re-examined the recently introduced conformal and SO(2) duality-rotation invariance NED model, given in Eq. (1). We applied the unitarity and casualty conditions to find an upper bound for the arbitrarily dimensionless constant γ in the theory. In accordance with our results its domain is given as 0 < γ < 0.8814. Furthermore, we minimally coupled this particular NED with Einstein's gravity. From the field equations, we obtained a Reissner-Nordstrom-type charged black hole solution with a new extra parameter, i.e., γ. Let's note that ω 2 = cosh γ − 1−q 2 1+q 2 sinh γ represents γ in our investigation. The effects of γ on the physical properties of the black hole solutions have been investigated. The thermal stability of the black hole, specifically, has been studied. The results have been demonstrated in Fig.  2 and Fig. 3. In accordance with Fig. 3, the stability of the black hole decreases as the value of the parameter γ increases.