Non-Commutative Correction to Thin Shell Collapse in Reissner Nordstr$\ddot{o}$m Geometry

This paper investigates the polytropic matter shell collapse in the non-commutative Reissner-Nordstr$\ddot{o}$m geometry. Using the Israel criteria, equation of motion for the polytropic matter shell is derived. In order to explore the physical aspects of this equation, the most general equation of state, $p=k{\rho}^{({1+\frac{1}{n}})}$, has been used for finite and infinite values of $n$. The effective potentials corresponding to the equation of motion have been used to explain different states of the matter shell collapse. The numerical solution of the equation of motion predicts collapse as well as expansion depending on the choice of initial data. Further, in order to include the non-commutative correction, we modify the matter components and re-formulate the equation of motion as well as the corresponding effective potentials by including non-commutative factor and charge parameter. It is concluded that charge reduces the velocity of the expanding or collapsing matter shell but does not bring the shell to static position. While the non-commutative factor with generic matter favors the formation of black hole.