Axially Symmetric Shear-free Fluids in $f(R,T)$ Gravity

In this work we have discussed the implications of shear-free condition on axially symmetric anisotropic gravitating objects in $f(R,T)$ theory. Restricted axial symmetry ignoring rotation and reflection enteries is taken into account for establishment of instability range. Implementation of linear perturbation on constitutive modified dynamical equations yield evolution equation. This equation associates adiabatic index $\Gamma$ with material and dark source components defining stable and unstable regions in Newtonian (N) and post-Newtonian (pN) approximations.