Dynamics and Stability via Thin-Shell of Approximated Black Holes in f ( Q ) $f(\mathbb {Q})$ Gravity

The current study is devoted to exploring the geometrical configuration of a thin-shell in the background of symmetric teleparallel gravity. For this purpose, the well-known cut-and-paste approach by matching the inner flat and outer newly calculated class of approximated black hole (BH) solutions in symmetric teleparallel gravity are considered, i.e., uncharged, charged, and anti-de-Sitter BHs. The dynamical analysis of thin-shell configuration by adopting the massive and massless scalar field via Klein-Gordon's equation of motion is discussed. The effective potential and proper time derivative of shell radius for both massive and massless scalar shells are used to discuss the collapse, expansion, and oscillatory behavior. The stable configuration of thin-shell is observed through the linearized radial perturbation approach with a phantomlike equation of state, i.e., quintessence, dark energy, and phantom energy. It is noted that stable/unstable behavior of thin-shell is found after the expected position of the event horizon of an exterior manifold. It is concluded that the stability of a thin-shell is greater for $f\left(Q\right)$ BH with cosmological constant as compared to the uncharged and charged $f\left(Q\right)$ BHs.