[Paper Review] Classification of radial Kerr geodesic motion

Geoffrey Compère, Yan Liu+1 more|arXiv (Cornell University)|Jun 6, 2021

Astrophysical Phenomena and Observations84 references31 citations

TL;DR

This paper presents a comprehensive classification of radial timelike geodesic motion in the exterior nonextremal Kerr spacetime by analyzing the root structures of the radial geodesic equation in terms of energy, angular momentum, and Carter's constant. It derives the complete phase space for all generic and nongeneric root systems—accounting for polar, time, and azimuthal motion constraints—identifying 11 distinct orbit classes, including bound, unbound, spherical, and marginal orbits, with explicit parametrization of the separatrix via double-root conditions.

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.

Motivation & Objective

• To systematically classify all possible radial geodesic motions in the exterior nonextremal Kerr spacetime based on the root structure of the radial potential.
• To determine the domain of existence for each root system in the phase space defined by energy, angular momentum, and Carter’s constant.
• To incorporate physical constraints from polar motion (via Carter’s constant), time and azimuthal motion (via ergoregion constraints), and horizon proximity to distinguish allowed from disallowed orbits.
• To derive the complete separatrix between generic orbit classes as the union of three distinct physical limits: pericenter double root, zero eccentricity, and turning point double root.
• To extend the classification to null geodesics and near-horizon extremal Kerr motion as limiting cases, comparing with existing literature.

Proposed method

• Uses a novel compact notation to classify inequivalent root structures of the first-order radial geodesic equation, treating roots as real or coincident.
• Applies constraints from polar motion to bound Carter’s constant Q, and from time/azimuthal motion to restrict allowed energy and angular momentum ranges within the ergoregion.
• Performs a taxonomy of 4 generic root structures (with simple roots) and 8 nongeneric cases (due to horizon coincidence, vanishing roots, or double roots).
• Derives explicit parametrization of the separatrix as the union of three regions: (1) pericenter double root in bound motion, (2) zero eccentricity limit, (3) turning point double root in unbound motion.
• Uses the discriminant of the quartic radial potential to identify critical values of energy, angular momentum, and Carter’s constant where root degeneracies occur.
• Treats null geodesics and near-horizon extremal Kerr motion as limiting cases of the general timelike classification.

Experimental results

Research questions

• RQ1 What are the complete set of inequivalent root structures for radial timelike geodesics in the Kerr spacetime, and how do they depend on energy, angular momentum, and Carter’s constant?
• RQ2 Which root systems correspond to physically allowed radial geodesic motions, given the constraints from polar, time, and azimuthal motion?
• RQ3 How is the separatrix between different orbit classes—specifically between bound and plunging, or bounded and unbounded—parametrized in terms of double-root conditions?
• RQ4 What are the distinct orbit classes in the ergoregion, especially for negative energy geodesics, and how do they differ from equatorial or non-ergoregion cases?
• RQ5 How do null geodesics and near-horizon extremal Kerr motion emerge as limiting cases of the general timelike classification?

Key findings

• The paper identifies 11 distinct geodesic orbit classes, including trapped, deflecting, plunging, spherical, and marginal orbits, based on root structure and physical constraints.
• For energy E = Eibco ≈ 1, the system exhibits a critical transition with k = kibco = 16M²μ², where a double root at r = 4M marks the separatrix between trapped and deflecting orbits.
• The separatrix is explicitly parametrized as the union of three regions: (1) pericenter double root in bound motion, (2) zero eccentricity limit, and (3) turning point double root in unbound motion, all described by the same quartic potential.
• For E > 1, the discriminant of the radial potential changes sign at k = ku > 16, leading to a transition from single-root (plunging) to three-root (bound or unbound) behavior, with unstable circular orbits Cu(E) at r = ru₂.
• In the ergoregion, six distinct values of Kerr angular momentum emerge as critical for radial motion with negative energy, reflecting the unique constraints on time and azimuthal motion.
• Null geodesics and near-horizon extremal Kerr motion are consistently recovered as limiting cases of the general timelike classification, validating consistency with prior literature.