Spherically symmetric and static solutions in $f(R)$ gravity coupled with EM fields

Abstract

Solutions of field equations in $f(R)$ gravity are found for a spherically symmetric and static spacetime in the Born-Infeld (BI) non-linear electrodynamics. It is found that the models supported in this configuration must have the parametric form $f^{\prime }(R){|}_r=m+nr$, with $m,n$ constants, whose value and sign have a strong impact on the solutions, as well as in the form and range of $f(R)$. When $n=0$, $f(R)=mR+m_0$ and the Einstein-BI solution is found. When $m\ne 0$ and $n\ne 0$, $f(R)$ is asymptotically equivalent to GR and the Schwarzschild and $f(R)$-Reissner-Nordström solutions are written in some limits, likewise if $n>0$ and $r\gg 1$, $f(R)$ can be found as a series approximation and as a particular case, when $R_S=-\frac{m^2}{3n}$, explicitly $f(R)=mR+2n\sqrt{R}+m_0$. Finally, the solutions, scalar curvature and parametric function $f(r)$ in the non-linear ($m=0$) regime of $f(R)$ are found, and some models for specific values of $m$ and $n$ are plotted.

14 pages, 2 figures