Impact of f(ℜ) functions on the dynamical evolution of axially symmetric collapsing stars

Abstract

The study explores the framework of $f\left(\Re \right)$ gravity and the dynamical instability of an axially symmetric collapsing star with anisotropic matter distribution by employing a specific equation of state that links the static and non-static components of physical quantities through the adiabatic index. In this modified gravity theory, the Lagrangian density depends on the Ricci scalar ($\Re$). We utilize radial perturbation techniques to analyze small variations in the geometric and material properties of the collapsing structure. The unstable regions are investigated using Newtonian and post-Newtonian approximations, with the collapse equation highlighting the critical role of the adiabatic index in identifying unstable phases of constrained, nonstatic, axially symmetric stars. Inequalities for the adiabatic index are derived for the Starobinsky model and logarithmic form of $f\left(\Re \right)$ gravity. In contrast, this modification introduces additional curvature terms that significantly affect the dynamics of considered compact formations. The findings indicate hydrostatic equilibrium is achieved through a balance among gravitational, anti-gravitational, and effective pressure forces, while an imbalance triggers instability. Furthermore, dark source terms arising due to considered modified gravity contribute to the unstable behavior of the structure during its evolution, influenced by the choice of specific $f\left(\Re \right)$ functions.