Causal Geodesics in Cylindrically Symmetric Vacuum Spacetimes Using Hamilton-Jacobi Formalism

Cylindrically symmetric vacuum spacetimes are of immense interest in theoretical physics due to its connection to cosmic strings hypothesized in quantum field theory. In this article, we explore the properties of such spacetime and provide the complete solution to the time- and light-like geodesics in it using Hamilton-Jacobi formalism. In addition, we compare several properties and massive particle trajectories of relativistic cylindrically symmetric vacuum spacetimes to its non-relativistic, Newtonian gravitational counterparts.