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

TL;DR

This paper classifies all static, spherically symmetric vacuum solutions with non-expanding principal null directions in $f(R)$ gravity, revealing multiple solutions beyond the Nariai spacetime and characterizing their geometric features.

Contribution

It identifies new vacuum solutions in $f(R)$ gravity with specific null direction properties, extending known solutions beyond General Relativity.

Findings

Multiple solutions for non-constant Ricci scalar in $f(R)$ gravity.

All solutions correspond to a specific $f(R)$ function involving the Ricci scalar.

The Nariai spacetime is not unique in this context.

Abstract

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 $f(R) = r_0^{-1}\left\lvert R-3/r_0^2\right\rvert^{1/2}$, where $r_0$ is a non-null constant. Finally, we briefly present some geometric properties of these solutions.