Evolution of Wormholes under f(R, T) Theory, the Karmarkar Condition and the Casimir Energy

Abstract: In this study, both the evolution of wormholes (by examining both the energy conditions and using the TOV equations) and the effects of the Karmarkar condition on the solutions obtained under certain specific cases were examined in the light of the $f(R,T)$ gravity theory, using two $f(R,T)$ functions predicted to describe the accelerated expansion of the universe. In this context, for the first time in the literature, a generalized shape function was obtained using the Karmarkar condition. It was observed that solutions of the type $R-a_{1}^2/R+a_{2}g(T)$ satisfy the energy conditions (with the dominant energy condition being partially satisfied), whereas solutions of the type $R+a_{1}^2R^2+a_{2}g(T)$ require the presence of exotic matter. In both cases, stable, static, and traversable wormhole solutions were obtained. By applying the Karmarkar condition to the $R+a_{1}^2R^2+a_{2}g(T)$ type solutions, which violate the energy conditions, the relationship between wormhole geometry and energy conditions was investigated. The study examined whether the Karmarkar condition eliminates the need for exotic matter, and it was found that the solutions do not remove the necessity of exotic matter. Additionally, it was demonstrated that a specific value of the parameter, ${\beta}$, which determines the radial variation of the shape function, could ensure the stability of the wormhole throat with the aid of Casimir energy. In other words, it is considered possible that the geometric evolution of the wormhole throat could trigger the transition from positive energy (baryonic matter) to negative energy (dark matter, dark energy, or other exotic matter) by inducing Casimir forces.