Static and spherically symmetric vacuum spacetimes with non-expanding principal null directions in $f(R)$ gravity

In this work we characterize all the static and spherically symmetric vacuum solutions in $f(R)$ gravity when the principal null directions of the Weyl tensor are non-expanding. In contrast to General Relativity, we show that the Nariai spacetime is not the only solution of this type when general $f(R)$ theories are considered. In particular, we find four different solutions for the non-constant Ricci scalar case, all of them corresponding to the same theory, given by $f(R)=r_0^{-1}{\mid R-3\mathrm{/}{r}_0^2\mid }^{1\mathrm{/}2}$, where $r_0$ is a non-null constant. Finally, we briefly present some geometric properties of these solutions.