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Dynamics of spinning test bodies in the Schwarzschild space-time: reduction and circular orbits

Ivan Bizyaev

TL;DR

This work delivers an explicit reduction of the Mathisson–Papapetrou–Tulczyjew system for a spinning test body in Schwarzschild spacetime to a 3-DOF Hamiltonian form on an invariant manifold, then further reduces to a 2-DOF system on that manifold to study circular orbits. It identifies both known symmetric circular orbits and new asymmetric ones, whose orbital angular momentum is not aligned with the body's spin, and analyzes their stability. A Poincaré map reveals successive pitchfork bifurcations and a heteroclinic tangle, signaling complex, potentially chaotic dynamics for sufficiently large spin, though the asymmetric branches are typically unstable for realistic compact objects. The results illuminate spin–curvature coupling effects beyond geodesic motion and point toward Kerr generalizations and broader dynamical implications in relativistic two-body problems.

Abstract

This paper investigates the motion of a rotating test body in the Schwarzschild space-time. After reduction, this problem reduces to an analysis of a three-degree-of-freedom. Hamiltonian system whose desired trajectories lie on the invariant manifold described by the Tulczyjew condition. An analysis is made of the fixed points of this system which describe the motion of the test body in a circle. New circular orbits are found for which the orbital angular momentum is not parallel to the angular momentum of the test body. Using a Poincare map, bifurcations of periodic solutions are analyzed.

Dynamics of spinning test bodies in the Schwarzschild space-time: reduction and circular orbits

TL;DR

This work delivers an explicit reduction of the Mathisson–Papapetrou–Tulczyjew system for a spinning test body in Schwarzschild spacetime to a 3-DOF Hamiltonian form on an invariant manifold, then further reduces to a 2-DOF system on that manifold to study circular orbits. It identifies both known symmetric circular orbits and new asymmetric ones, whose orbital angular momentum is not aligned with the body's spin, and analyzes their stability. A Poincaré map reveals successive pitchfork bifurcations and a heteroclinic tangle, signaling complex, potentially chaotic dynamics for sufficiently large spin, though the asymmetric branches are typically unstable for realistic compact objects. The results illuminate spin–curvature coupling effects beyond geodesic motion and point toward Kerr generalizations and broader dynamical implications in relativistic two-body problems.

Abstract

This paper investigates the motion of a rotating test body in the Schwarzschild space-time. After reduction, this problem reduces to an analysis of a three-degree-of-freedom. Hamiltonian system whose desired trajectories lie on the invariant manifold described by the Tulczyjew condition. An analysis is made of the fixed points of this system which describe the motion of the test body in a circle. New circular orbits are found for which the orbital angular momentum is not parallel to the angular momentum of the test body. Using a Poincare map, bifurcations of periodic solutions are analyzed.

Paper Structure

This paper contains 16 sections, 4 theorems, 148 equations, 13 figures.

Table of Contents

  1. Introduction
  2. The Mathisson --- Papapetrou equations
  3. Newtonian gravitation
  4. Relativistic theory
  5. The Tulczyjew condition
  6. Hamiltonian form
  7. The Schwarzschild space-time
  8. Additional integrals
  9. Reduction in the case $\vec{Q}\neq 0$
  10. Reduction in the case $\vec{Q} = 0$
  11. Relative equilibria
  12. Symmetric circular orbits
  13. Asymmetric circular orbits
  14. Stability analysis
  15. Poincaré map
  16. ...and 1 more sections

Key Result

Proposition 1

Let $p^\alpha$ be the linear momentum of the test body and let $S^{\alpha \beta}$ be its tensor of angular momentum, which are defined by relations Eq_PS. Their evolution in the pole-dipole approximation is governed by the following system of equations: where $R^{\alpha}_{ \ \mu \nu \lambda}$ is the curvature tensor and the following covariant derivative has been introduced: where $\Gamma^\alpha

Figures (13)

  • Figure 1: A test body.
  • Figure 2: A schematic representation of the world tube.
  • Figure 3: Curves $\Sigma_0^\pm$ on the plane $(\ell, \mathcal{E})$ for different $c$.
  • Figure 4: Domain of definition (gray) of the radial coordinate $r$ depending on $c$ for $\Sigma^{\pm}_0$.
  • Figure 5: Region on the plane $(z, \mathcal{E})$ and $(c, \mathcal{E})$ for which there exist equilibrium points $\mathcal{N}_1^\pm$ (gray). Black indicates the curve on which $E_3^\pm=0$.
  • ...and 8 more figures

Theorems & Definitions (12)

  • Proposition 1
  • Remark 1
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Remark 2
  • Remark 3
  • Proposition 1
  • Remark 4
  • ...and 2 more