This paper analyzes the sustenance of a traversable wormhole under the most optimistic assumptions, including the violation of the Null Energy Condition, a finite redshift function, static and symmetric geometry, and the satisfaction of the flare-out condition. This study also examines the Morris-Thorne Metric to understand the behavior of the geometry of a traversable wormhole. We investigate the response of the wormhole to perturbations induced in the geometry. The analysis demonstrates that the wormhole geometry is highly sensitive to small perturbations. We derive a general relation for perturbation in the throat radius δr0 in terms of a definite integral, demonstrating that perturbations in the throat radius depend on the cumulative distribution of energy density and, therefore, are non-local. The result further demonstrates a non-local coupling between geometry and stress-energy configuration at the throat. The definite integral solved for a specific shape function of choice yields the relation, δr0 = (1/2) δb (r0) }, which demonstrates that the perturbation in the throat directly influences its geometry. We derive another general relation of δr0 = δb (r0) / (1 - b' (r0) ). In this relation, we analyze how changes in (1 - b' (r0) ) ^ (-1) amplify the perturbation at the throat. These results indicate that maintaining a stabilized geometry requires precise coordination between geometry and stress-energy. Consequently, even under idealized conditions, sustaining a traversable wormhole appears to be extremely difficult.
Aayush Thakur (Mon,) studied this question.