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

TL;DR

This work analyzes static plane-symmetric configurations within $f(Q)$ gravity, uncovering that vacuum regions enforce a constant nonmetricity scalar $Q=Q_0$ which yields Taub-$( ext{A)dS}$ spacetimes with a cosmological constant $oxed{\Lambda = -rac{Q_0}{2} ext{}}$. It then studies how singular thin shells and finite-thickness slabs can source these vacua, deriving junction conditions that connect shell densities and interior pressures to exterior integration constants. A numerical exploration of the quadratic model $f(Q)=Q+oldsymbol{ ext{ aisebox{0.2ex}{-}} } Q^2$ shows that $Q_0=-rac{1}{3oldsymbol{ ext{ aisebox{0.2ex}{-}} }}$, with negative $oldsymbol{ ext{ aisebox{0.2ex}{-}} }$ producing thicker slabs and larger central pressures, while positive $oldsymbol{ ext{ aisebox{0.2ex}{-}} }$ cannot realize two natural surfaces. These results illustrate how planar symmetry constrains vacuum branches and interior structures in $f(Q)$ gravity and inform the viability of specific $f(Q)$ models in symmetric spacetimes, with implications for modeling domain walls and related configurations.

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 $f(Q)=Q+αQ^2$ with isotropic matter, we show that the maximum pressure inside the slab generally does not coincide with the geometric center. Moreover, a negative $α$ with larger magnitude leads to higher internal pressure and a thicker slab, while models with positive $α$ are incompatible with a self-gravitating slab of positive pressure.

Figures (2)

• Figure 1: The profiles of the isotropic pressure $p$ for $\alpha=-0.1\rho_0^{-1}$ and various $u'(z_-)$. The peaks locate at $(-0.004,0.0015)$, $(-0.008,0.028)$, $(-0.024,0.089)$, $(-0.056,0.215)$, $(-0.081,0.320)$, for $u_0'/\sqrt{\rho_0}=-0.1$, $-0.5$, $-1$, $-2$, $-3$, respectively, where the geometric center $z=(z_-+z_+)/2$ has been moved to $z=0$.
• Figure 2: The profiles of the isotropic pressure $p$ for $u'(z_-)=-0.1\sqrt{\rho_0}$ and various $\alpha$. The peaks locate at $(-0.0008,0.006)$, $(-0.005,0.020)$, $(-0.008,0.029)$, $(-0.010,0.046)$, for $\alpha\rho_0=-0.01$, $-0.05$, $-0.1$, $-0.5$, where the geometric center $z=(z_-+z_+)/2$ has been moved to $z=0$.