Geodesics on metrics of self-dual Taub-Nut type

TL;DR

This paper analyzes geodesic equations on self-dual Taub-NUT type metrics, providing explicit solutions under certain conditions and highlighting unresolved cases for complete understanding of particle trajectories.

Contribution

It offers explicit solutions for geodesics when multiple coordinates are constant and identifies cases where solutions are still unknown, advancing understanding of these geometries.

Findings

Explicit solutions for geodesics with multiple constants of motion

Identification of cases with unresolved geodesic equations

Enhanced understanding of self-dual Taub-NUT metrics

Abstract

Geodesic equations are solved when at least two of $\tau$, $\theta$, $\varphi$ are constant on metrics of self-dual Taub-NUT type. They can also be solved also on self-dual Taub-NUT metrics if only $r$, $\theta$ or $\varphi$ is constant. However, the explicit solution of the geodesic equations is not available yet if only $\tau$ is constant.