Stability Analysis via Local Density Perturbations: Cracking and Overturning in Modified Gravity

This paper examines how local density perturbations affect the stability of anisotropic stars in the f(𝒬L m ) gravity, with 𝒬 denoting the non-metricity scalar and L m is the matter Lagrangian. The field equations for this theory are introduced and the associated condition for hydrostatic equilibrium is derived. Further, the linear model of this theory is used to obtain explicit equations. The static spherical symmetry is analyzed by a non-singular solution, whereas its metric constants are fixed via matching condition. We develop a systematic framework to detect the cracking and the overturning instabilities by examining sign transition in the net force per unit density. The stability of anisotropic stars is analyzed graphically by applying local density perturbations to the hydrostatic equilibrium. Our findings provide important insights into the structural stability of stars and confirm that the cracking and overturning method serve as a reliable diagnostic for stability analysis within the modified gravity.