Impact of Radial Perturbations on Expansion-free Anisotropic Fluid Spheres in D-dimensional Modified Gravity

In this manuscript, we have continued the work of Herrera et al. (Gen. Relativ. Gravit. 44, 1143, 2012) to investigate the instability of non-static spheres supported by anisotropic matter configuration in the background of D-dimensional Einstein gravity. For this purpose, we assume that our relativistic sphere undergoes adiabatic evolution with a vanishing expansion scalar condition. We compute gravitational field equations, conservation equations, and junction conditions in D-dimensional Einstein theory. After considering the evolution of the systems under an expansion-free background, some significant constraints are established by using a perturbation scheme. This leads us to develop the dynamical instability of expansion-free anisotropic fluid spheres in the Newtonian (N) and Post-Newtonian (PN) domains. After focusing on the N and PN eras, our study reveals that the adiabatic index ( \(\Gamma \) $\mathrm{\Gamma }$ ), does not contribute to the instability ranges in these domains. Instead, these ranges are being determined solely by the anisotropy of fluid pressure, D-dimensional Einstein gravity parameter, and the radial energy density profile.