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

TL;DR

This paper develops a comprehensive method to determine all possible geodesic paths in axially symmetric space-times, including Kerr and Schwarzschild metrics, aiding the understanding of orbits around compact objects.

Contribution

It introduces a method to find all geodesics in axially symmetric space-times and derives the most general Killing tensor and constant of motion.

Findings

Explicit solutions for Kerr geodesics

Recovery of Schwarzschild geodesics in the static limit

Identification of the most general Killing tensor

Abstract

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.