Static plane symmetric solutions in $f(Q)$ gravity

TL;DR

This paper explores static plane symmetric solutions in $f(Q)$ gravity, deriving vacuum solutions akin to Taub-(anti) de Sitter spacetimes and analyzing matter sources like thin shells and slabs.

Contribution

It provides a systematic derivation of vacuum solutions in $f(Q)$ gravity and examines matter configurations, including numerical analysis of a quadratic $f(Q)$ model.

Findings

Vacuum solutions correspond to Taub-(anti) de Sitter spacetimes with a cosmological constant.

Maximum pressure in slabs does not occur at the geometric center.

Negative $eta$ in quadratic models increases internal pressure and slab thickness.

Abstract

We systematically investigate static plane symmetric configurations in $f(Q)$ gravity. For vacuum regions, we discuss the constancy of the nonmetricity scalar $Q$ and derive general vacuum solutions, which correspond effectively to Taub-(anti) de Sitter spacetimes with a cosmological constant determined by the specific $f(Q)$ model. By matching a singular thin shell source to the vacuum solutions, we relate the shell's energy density and pressure to the integration constants of the exterior geometry. We also examine a finite-thickness slab as another matter source supporting the vacuum solution. Through numerical analysis of a quadratic model $f(Q)=Q+\alpha Q^2$ with isotropic matter, we show that the maximum pressure inside the slab generally does not coincide with the geometric center. Moreover, a negative $\alpha$ with larger magnitude leads to higher internal pressure and a thicker slab, while models with positive $\alpha$ are incompatible with a self-gravitating slab of positive pressure.