Classification of radial Kerr geodesic motion

TL;DR

This paper provides a comprehensive classification of radial geodesic motion in Kerr spacetime by analyzing root structures of the radial equation, including generic and non-generic cases, with explicit phase space parametrization and extremal limits.

Contribution

It introduces a novel taxonomy of root structures for Kerr radial geodesics, including explicit phase space and constraints, extending previous classifications to non-generic cases and extremal limits.

Findings

Classified all root structures of Kerr radial geodesics.

Derived explicit phase space parametrization in terms of physical constants.

Compared null and extremal Kerr cases with existing literature.

Abstract

We classify radial timelike geodesic motion of the exterior non-extremal Kerr spacetime by performing a taxonomy of inequivalent root structures of the first order radial geodesic equation using a novel compact notation and by implementing the constraints from polar, time and azimuthal motion. Four generic root structures with only simple roots give rise to eight non-generic root structures when either one root becomes coincident with the horizon, one root vanishes or two roots becomes coincident. We derive the explicit phase space of all such root systems in the basis of energy, angular momentum and Carter's constant and classify whether each corresponding radial geodesic motion is allowed or disallowed from existence of polar, time and azimuthal motion. The classification of radial motion within the ergoregion for both positive and negative energies leads to 6 distinguished values of the Kerr angular momentum. The classification of null radial motion and near-horizon extremal Kerr radial motion are obtained as limiting cases and compared with the literature. We explicitly parametrize the separatrix describing root systems with double roots as the union of the following three regions that are described by the same quartic respectively obtained when (1) the pericenter of bound motion becomes a double root; (2) the eccentricity of bound motion becomes zero; (3) the turning point of unbound motion becomes a double root.