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

TL;DR

This paper analytically solves the geodesic equations in Taub-NUT space-times using elliptic functions, analyzing particle motion, geodesic completeness, and extensions, with implications for understanding the space-time's structure.

Contribution

It provides the complete analytical solutions of geodesics in Taub-NUT space-times and examines their properties and extensions, addressing longstanding issues of geodesic completeness.

Findings

Analytical solutions expressed via Weierstrass elliptic functions.

Characterization of geodesic motion through polynomial zeros.

Discussion of geodesic completeness and extensions in Taub-NUT space-times.

Abstract

We present the complete set of analytical solutions of the geodesic equation in Taub-NUT space-times in terms of the Weierstrass elliptic function. 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, and the analytic extension of Miller, Kruskal and Godfrey, and compare with the Reissner-Nordstr\"om space-time.