Locally Rotationally Symmetric spacetimes of type II in $f(\mathcal{Q})$ gravity

Abstract

We investigate the $1+1+2$ covariant formalism in the presence of nonmetricity. Focusing on Locally Rotationally Symmetric spacetimes, we show how nonmetricity affects all the kinematic quantities involved in the covariant $1+1+2$ decomposition. We apply the resulting geometrical framework to study both homogeneous solutions and static spherically symmetric solutions in the context of $f(\mathcal{Q})$ gravity. We obtain sufficient conditions for homogeneous solutions with flat spatial hypersurfaces and Schwarzschild-de Sitter type solutions in the $1+1+2$ formalism. We also explore an elementary gravastar solution utilizing covariant junction conditions.