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Symplectic mechanics of relativistic spinning compact bodies I.: Covariant foundations and integrability around black holes

Paul Ramond

Abstract

In general relativity, the motion of an extended body moving in a given spacetime can be described by a particle on a (generally non-geodesic) worldline. In first approximation, this worldline is a geodesic of the underlying spacetime, and the resulting dynamics admit a covariant and 4-dimensional Hamiltonian formulation. In the case of a Kerr background spacetime, the Hamiltonian was shown to be integrable by B.~Carter and the now eponymous constant. At the next level of approximation, the particle possesses proper rotation (hereafter \textit{spin}), which couples the curvature of spacetime and drives the representative worldline away from geodesics. In this article, we lay the theoretical foundations of a series of works aiming at exploiting the Hamiltonian nature of the equations governing the motion of a spinning particle, at linear order in spin. Our formalism is covariant and 10-dimensional. It handles the degeneracies inherent to the local Lorentz invariance of general relativity with tools from Poisson geometry, and accounts for the center-of-mass/spin-supplementary-condition using constrained Hamiltonian system theory. As a first application, we consider the linear-in-spin motion in a Kerr background. We show that the resulting Hamiltonian system admits exactly five functionally independent integrals of motion related to Killing symmetries, thereby proving that the system is integrable. We conclude that linear-in-spin corrections to the geodesic motion do not break integrability, and that the resulting trajectories are not chaotic. We explain how this integrability feature can be used to reduce the computational cost of waveform generation schemes for asymmetric binary systems of compact objects.

Symplectic mechanics of relativistic spinning compact bodies I.: Covariant foundations and integrability around black holes

Abstract

In general relativity, the motion of an extended body moving in a given spacetime can be described by a particle on a (generally non-geodesic) worldline. In first approximation, this worldline is a geodesic of the underlying spacetime, and the resulting dynamics admit a covariant and 4-dimensional Hamiltonian formulation. In the case of a Kerr background spacetime, the Hamiltonian was shown to be integrable by B.~Carter and the now eponymous constant. At the next level of approximation, the particle possesses proper rotation (hereafter \textit{spin}), which couples the curvature of spacetime and drives the representative worldline away from geodesics. In this article, we lay the theoretical foundations of a series of works aiming at exploiting the Hamiltonian nature of the equations governing the motion of a spinning particle, at linear order in spin. Our formalism is covariant and 10-dimensional. It handles the degeneracies inherent to the local Lorentz invariance of general relativity with tools from Poisson geometry, and accounts for the center-of-mass/spin-supplementary-condition using constrained Hamiltonian system theory. As a first application, we consider the linear-in-spin motion in a Kerr background. We show that the resulting Hamiltonian system admits exactly five functionally independent integrals of motion related to Killing symmetries, thereby proving that the system is integrable. We conclude that linear-in-spin corrections to the geodesic motion do not break integrability, and that the resulting trajectories are not chaotic. We explain how this integrability feature can be used to reduce the computational cost of waveform generation schemes for asymmetric binary systems of compact objects.
Paper Structure (63 sections, 106 equations, 3 figures, 2 tables)

This paper contains 63 sections, 106 equations, 3 figures, 2 tables.

Table of Contents

  1. The relativistic motion of test bodies
  2. Relativistic Hamiltonian formulation
  3. The question of integrability around black holes
  4. Motivation and main results
  5. Organization of the paper
  6. Spinning point particles in general relativity
  7. The dipolar MPTD system
  8. Spin supplementary conditions
  9. The Tulzyjew-Dixon SSC
  10. Linear-in-spin dynamics under the TD SSC
  11. The MPTD equations as a constrained Poisson system
  12. Definition of the Poisson system
  13. Phase space
  14. Poisson structure
  15. Hamiltonian function
  16. ...and 48 more sections

Figures (3)

  • Figure 1: The open domain of definition $\Delta\subset\mathbb{R}^2$ for the canonical spin momenta $({\pi_\sigma},{\pi_\zeta})$.
  • Figure 2: Physical interpretation of the symplectic variables $(\sigma,{\pi_\sigma},\zeta,{\pi_\zeta})$ in the 3-dimensional euclidean subspace $(\vec{e}_1,\vec{e}_2,\vec{e}_3)$.
  • Figure 3: Different phase space sub-manifold are necessary to lift the degeneracies associated to the existence of Casimir invariants and correctly implementing the TD SSC.