Exact solutions of $f(R)$ gravity coupled to nonlinear electrodynamics

In this work, exact solutions of static and spherically symmetric space-times are analyzed in $f(R)$ modified theories of gravity coupled to nonlinear electrodynamics. First, we restrict the metric fields to one degree of freedom, considering the specific case of $g_{tt}{g}_{rr}=-1$. Using the dual $P$ formalism of nonlinear electrodynamics, an exact general solution is deduced in terms of the structural function $H_P$. In particular, specific exact solutions to the gravitational field equations are found, confirming previous results and new pure electric field solutions are found. Second, motivated by the existence of regular electric fields at the center, and allowing for the case of $g_{tt}{g}_{rr}\ne -1$, new specific solutions are found. Finally, we outline alternative approaches by considering the specific case of constant curvature, followed by the analysis of a specific form for $f(R)$.

Figure 1

In the left panel we plot $R(r)$ and $f(r)$, while the right panel shows $f(R)$. We choose the coupling constants to be $B=1=q_e$ and the overall additive constant $C_1=0$. The sign of the constant $D$ influences the sign of $R$ and $f$ close to the center. For the choice $C_2=2$ it reduces to $D=2(A-1)$ and by plotting $f$ and $R$ for $A\in \{0.5,1,1.2\}$ we see the different behaviors: as soon as $D$ is positive (here $A>1$) $f(R)$ is not a uniquely defined function at large distances $r$ from the center.

Figure 2

In the left panel we plot $p_{\pm }$, the powers of $r$ in the solution for $f_R(r)$, as function of the parameter $\ell$. The right panel zooms into the right branch of $p_{\pm }$. Note that $p_{-}$ vanishes for $\ell =0$, is positive for negative $\ell$ and vice versa, while $p_{-}\to -2$ for large $\ell$. This behavior makes the solution $f_R=B{r}^{p_{-}}$ particularly interesting.

Figure 3

In the left panel the exponents of $r$ in the four terms of $e^{2\alpha (r)}$ for $\ell \ne 1,2$ are plotted. For $\ell \ge {\ell }_{\mathrm{crit}}\simeq 0.7$ the exponents of $r$ are positive in all terms. Negative $\ell$ lead to solutions divergent at the center. At $\ell =1$ and $\ell =2$, the hierarchy of the terms changes, which explains why these are special cases. The right panel shows $e^{2\alpha (r)}$ for $\ell =0.8$ for four different combinations of signs of the integration constants, while for simplicity and transparency we set $K=1=k^2$ throughout the analysis.