Regular multihorizon black holes in $f\mathbf{(}G\mathbf{)}$ gravity with nonlinear electrodynamics

In this work, we study the existence of regular black hole solutions with multihorizons in general relativity and in some alternative theories of gravity. We consider the coupling between the gravitational theory and nonlinear electrodynamics. The coupling generates modifications in the electromagnetic sector. The main objective of this paper is to generalize the solutions already known from general relativity to $f(G)$ theory. First, we correct some misprints of Odintsov and Nojiri’s work in order to introduce the formalism that is used in $f(G)$ gravity. In order to satisfy all field equations, the method to find solutions to alternative theories generates different $f(R)$ and $f(G)$ functions for each solution, where only the nonlinear term of $f(G)$ contributes to the field equations. We also analyze the energy conditions, since it is expected that some must be violated to find regular black holes, and using an auxiliary field, we analyze the nonlinearity of the electromagnetic theory.

Figure 1

Electromagnetic Lagrangian for the asymptotically flat solution with $\beta =(4/3)m^3$ and $\alpha =(7/3)m$.

Figure 2

Graphical representation of $L(I)\times I$ associated with the asymptotically anti–de Sitter solution with $r_1=0.5m$ and $r_2=1.5m$.

Figure 3

Behavior of the electromagnetic Lagrangian in terms of $I$ associated with the solution with three horizons for $r_1=0.5m$, $r_2=1.5m$, $r_3=16m$, and $l=20m$.

Figure 4

Comparison between the $f(G)$ function, that generates the solution (13), and a linear term of $G$ with $q=0.1m$.

Figure 5

Behavior of the electromagnetic fluid quantities in terms of the radial coordinate with $c_1=1m$ and $q=0.1m$.

Figure 6

Parameters of the equations of state in terms of the radial coordinate with $q=0.1m$.