Analytic treatment of complete and incomplete geodesics in Taub-NUT space-times

We present the complete set of analytical solutions of the geodesic equation in Taub-NUT space-times in terms of the Weierstrass elliptic functions. We systematically study the underlying polynomials and characterize the motion of test particles by its zeros. Since the presence of the Misner string'' in the Taub-NUT metric has led to different interpretations, we consider these in terms of the geodesics of the space-time. In particular, we address the geodesic incompleteness at the horizons discussed by Misner and Taub [C. W. Misner and A. H. Taub, Sov. Phys. JETP 28, 122 (1969) [Zh. Eksp. Teor. Fiz. 55, 233 (1968)]], and the analytic extension of Miller, Kruskal and Godfrey [J.G. Miller, M. D. Kruskal, and B. Godfrey, Phys. Rev. D 4, 2945 (1971)], and compare with the Reissner-Nordstrom space-time.