Super-extremal black holes in the quasitopological electromagnetic field theory

It has recently been proved that a simple generalization of electromagnetism, referred to as quasitopological electromagnetic field theory, is characterized by the presence of dyonic black-hole solutions of the Einstein field equations that, in certain parameter regions, are characterized by four horizons. In the present compact paper we reveal the existence, in this non-linear electrodynamic field theory, of super-extremal black-hole spacetimes that are characterized by the four degenerate functional relations $[g_{00}(r)]_{r=r_{\text{H}}}=[dg_{00}(r)/dr]_{r=r_{\text{H}}}=[d^2g_{00}(r)/dr^2]_ {r=r_{\text{H}}}=[d^3g_{00}(r)/dr^3]_{r=r_{\text{H}}}=0$, where $g_{00}(r)$ is the $tt$-component of the curved line element and $r_{\text{H}}$ is the black-hole horizon radius. In particular, using analytical techniques we prove that the quartically degenerate super-extremal black holes are characterized by the universal (parameter-{\it independent}) dimensionless compactness parameter $M/r_{\text{H}}={2\over3}(2\gamma+1)$, where $\gamma\equiv{_2F_1}(1/4,1;5/4;-3)$.


I. INTRODUCTION
The canonical Reissner-Nordström solution [1] of the coupled Einstein-Maxwell field equations describes a twoparameter family of asymptotically flat black-hole spacetimes.These curved spacetimes are characterized by two conserved (asymptotically measured) physical parameters: the mass M and the electric charge Q of the central black hole.
In standard Maxwell electrodynamics the boundary between black-hole spacetimes and horizonless naked singularities is marked by the presence of a zero-temperature extremal black hole whose horizon radius r = r H is characterized by the doubly degenerate functional relations where the radially-dependent metric function of the curved spacetime is given by the expression g 00 (r) = 1 − 2M/r + Q 2 /r 2 .From (1) one finds that the extremal Reissner-Nordström black hole is characterized by the familiar dimensionless relations [2] Recently, a physically intriguing (and mathematically elegant) generalization of Maxwell's electromagnetism, referred to as quasitopological electromagnetic field theory, has been studied in [3] (see also [4][5][6] and references therein).As discussed in [3], since the effects of the generalized non-linear quasitopological electromagnetic field theory are not manifest in Earth-based experiments, it may provide a reasonable phenomenologically rich alternative to the standard Maxwell field theory.
Intriguingly, it has been shown in [3] that, when coupled to gravity, the newly suggested quasitopological electromagnetic field theory is characterized by the presence of asymptotically flat dyonic black-hole spacetimes that, as opposed to the standard Reissner-Nordström black-hole solutions of the Einstein-Maxwell theory, may have four horizons.
The main goal of the present compact paper is to reveal the physically interesting fact that the non-linear electrodynamic field theory [3] is characterized by the presence of a unique family of super-extremal black-hole spacetimes whose horizons are quartically degenerate.In particular, below we shall use analytical techniques in order to determine the physical parameters that characterize these unique super-extremal black holes of the non-trivial (non-linear) electromagnetic field theory.

II. SUPER-EXTREMAL BLACK HOLES IN THE QUASITOPOLOGICAL NON-LINEAR ELECTROMAGNETIC FIELD THEORY
The quasitopological electromagnetic field theory is characterized by the composed action [3] where and {α 1 , α 2 } are the coupling parameters of the non-linear field theory [7].Note that the standard Maxwell theory is characterized by the simple relations α 1 = 1 and α 2 = 0.
Assuming a spherically symmetric curved spacetime of the form [1, 8] one finds that the quasitopological electromagnetic field theory (3), when coupled to the Einstein equations, is characterized by the presence of dyonic black-hole solutions of mass M , electric charge q, and magnetic charge p/α 1 of the form [3] where 2 F 1 (a, b; c; z) is the hypergeometric function [9].As shown in [3], for this non-vacuum spacetime, the null energy condition, the weak energy condition, and the dominant energy condition are all respected in the regime α 1 > 0 with α 2 > 0, whereas the strong energy condition is violated.Before proceeding, it is worth stressing the fact that the presence of the hypergeometric function 2 F 1 (a, b; c; z) in the curved line element (5) makes it a highly non-trivial task to explore, using purely analytical techniques, the physical and mathematical properties of these dyonic black-hole spacetimes.Nevertheless, we shall now prove explicitly that some interesting features of the recently proposed non-linear electrodynamic field theory (3) can be deduced from a direct inspection of the non-trivial curved line element (5).
In particular, we shall prove that the quasitopological electromagnetic field theory (3) is characterized by the presence of super-extremal black-hole spacetimes which are characterized by the quartically degenerate functional relations [Compare (6) with the doubly degenerate relations (1) that characterize the extremal black-hole solutions of the standard Maxwell theory.]In particular, using analytical techniques we shall determine the physical parameters that characterize the super-extremal black holes of the quasitopological non-linear electromagnetic field theory (3).