Geometry of the straight spinning string space-time

Abstract This work investigates the geometric and dynamical structure of a spacetime governed by the straight spinning string (SSS) metric. The analysis begins with a comprehensive study of the spacetime geometry, including computation of Christoffel symbols, Ricci tensor, and curvature forms in both coordinate and Cartan formalisms. Despite nontrivial global features such as frame-dragging and conical singularities, the spacetime is shown to be locally flat and Ricci-flat. The symmetries of geodesic motion are then explored via the Noether symmetry approach, yielding conserved quantities associated with the metric's isometries. A Hamiltonian formulation is subsequently developed on a four-dimensional configuration space, incorporating the effects of frame-dragging through canonical momenta and enabling a symplectic reduction. The resulting reduced Hamiltonian reveals hidden integrals linked to dynamical and geometric symmetries. The study proceeds to derive an effective potential for radial geodesic motion, elucidating the interplay between geometry, conserved charges, and singularities. Finally, symbolic and numerical analyses of the potential's behavior uncover a smooth profile near critical radii and offer insights into geodesic confinement and causal structure. Altogether, the results contribute to a deeper understanding of relativistic systems with angular-temporal couplings and topological defects. All calculations, numerical evaluations, and plots of the effective potential were performed using Maple 2020 to ensure accuracy and clarity in visualization.