Static plane symmetric solutions in f(Q) gravity

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 $$f(Q)=Q+\alpha Q^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>Q</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:mi>Q</mml:mi> <mml:mo>+</mml:mo> <mml:mi>α</mml:mi> <mml:msup> <mml:mi>Q</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:math> with isotropic matter, we show that the maximum pressure inside the slab generally does not coincide with the geometric center. Moreover, a negative $$\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> with larger magnitude leads to higher internal pressure and a thicker slab, while models with positive $$\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> are incompatible with a self-gravitating slab of positive pressure.