All analytic solutions for geodesic motion in axially symmetric space-times

Recent observations of the orbits of star clusters around Sgr \(A^\star\), imaging of black holes and gravitational waveforms of merging compact objects require a detailed understanding of the general relativistic geodesic motion. We came up with a method to provide all the possible geodesics in an axially symmetric space-time. The Kerr metric is explicitly worked out, recovering the Schwarzschild geodesics in the static limit. We also found the most general Killing tensor and its associated constant of motion for an axisymmetric space-time. The relevance of these results is crucial to understanding the different scenarios and the fundamental nature of the compact object at the galactic center.