Higher Order Corrections to Antisymmetric Spherical Solutions in $f(R)$ gravity

The static, spherical solutions that exhibit an antisymmetry between the temporal and radial coordinates in $f(R)$ gravity theories are presented. I present the constraint for this antisymmetry and show that pure $R^2$ models produce these solutions. An asymptotic expansion approach is taken to find the solutions, presenting first and second order corrections to the Schwarzschild metric. The null geodesics and photon sphere are determined for these corrections. Furthermore, I show that electromagnetic fields produce this antisymmetry as well. The metric solutions are analyzed in the context of black holes, bringing forward potentially a new class of black holes in modified gravity theories.