A Study of Levia-Civita and Cosmic String Solutions in Modified $f(R)$ Gravity

In this research manuscript, we explore cylindrically symmetric solutions within the framework of modified $f(R)$ theories of gravity, where $R$ representing the Ricci scalar. The study focuses on analyzing the cylindrical solutions within realistic space-time regions and investigates three distinct cases of exact solutions derived from the field equations of $f(R)$ theory of gravity. Moreover, we examine well-known Levi-Civita and cosmic string solutions for the said modify gravity. Furthermore, the manuscript delves into the implications of the obtained results by exploring energy conditions for each case. Notably, a violation of null energy conditions is observed, suggesting the potential existence of cylindrical wormholes. The findings contribute to the understanding of modified $f(R)$ theories of gravity, providing valuable insights into the nature of cylindrically symmetric solutions and their implications in theoretical physics.