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Canonical Hamiltonian for an extended test body in curved spacetime: To quadratic order in spin

Justin Vines, Daniela Kunst, Jan Steinhoff, Tanja Hinderer

TL;DR

This work constructs a canonical Hamiltonian for an extended spinning test body in curved spacetime, valid to quadratic order in spin, using three-dimensional canonical variables and explicit SSC handling. By formulating constrained actions for generic SSCs and performing a manifestly covariant worldline shift via bitensor methods, the authors obtain a Hamiltonian in the Newton–Wigner frame and then specialize to Kerr spacetime to yield a fully relativistic spinning test-body Hamiltonian, including spin-induced quadrupole effects. The Hamiltonian is expanded in Kerr spin and PN orders, and the results are shown to reproduce the test-mass limits of high-order PN spin couplings, providing a strong consistency check and valuable inputs for effective-one-body waveform models. Overall, the framework offers a rigorous route to incorporate finite-size and spin effects into relativistic two-body dynamics and gravitational-wave modeling.

Abstract

We derive a Hamiltonian for an extended spinning test body in a curved background spacetime, to quadratic order in the spin, in terms of three-dimensional position, momentum, and spin variables having canonical Poisson brackets. This requires a careful analysis of how changes of the spin supplementary condition are related to shifts of the body's representative worldline and transformations of the body's multipole moments, and we employ bitensor calculus for a precise framing of this analysis. We apply the result to the case of the Kerr spacetime and thereby compute an explicit canonical Hamiltonian for the test-body limit of the spinning two-body problem in general relativity, valid for generic orbits and spin orientations, to quadratic order in the test spin. This fully relativistic Hamiltonian is then expanded in post-Newtonian orders and in powers of the Kerr spin parameter, allowing comparisons with the test-mass limits of available post-Newtonian results. Both the fully relativistic Hamiltonian and the results of its expansion can inform the construction of waveform models, especially effective-one-body models, for the analysis of gravitational waves from compact binaries.

Canonical Hamiltonian for an extended test body in curved spacetime: To quadratic order in spin

TL;DR

This work constructs a canonical Hamiltonian for an extended spinning test body in curved spacetime, valid to quadratic order in spin, using three-dimensional canonical variables and explicit SSC handling. By formulating constrained actions for generic SSCs and performing a manifestly covariant worldline shift via bitensor methods, the authors obtain a Hamiltonian in the Newton–Wigner frame and then specialize to Kerr spacetime to yield a fully relativistic spinning test-body Hamiltonian, including spin-induced quadrupole effects. The Hamiltonian is expanded in Kerr spin and PN orders, and the results are shown to reproduce the test-mass limits of high-order PN spin couplings, providing a strong consistency check and valuable inputs for effective-one-body waveform models. Overall, the framework offers a rigorous route to incorporate finite-size and spin effects into relativistic two-body dynamics and gravitational-wave modeling.

Abstract

We derive a Hamiltonian for an extended spinning test body in a curved background spacetime, to quadratic order in the spin, in terms of three-dimensional position, momentum, and spin variables having canonical Poisson brackets. This requires a careful analysis of how changes of the spin supplementary condition are related to shifts of the body's representative worldline and transformations of the body's multipole moments, and we employ bitensor calculus for a precise framing of this analysis. We apply the result to the case of the Kerr spacetime and thereby compute an explicit canonical Hamiltonian for the test-body limit of the spinning two-body problem in general relativity, valid for generic orbits and spin orientations, to quadratic order in the test spin. This fully relativistic Hamiltonian is then expanded in post-Newtonian orders and in powers of the Kerr spin parameter, allowing comparisons with the test-mass limits of available post-Newtonian results. Both the fully relativistic Hamiltonian and the results of its expansion can inform the construction of waveform models, especially effective-one-body models, for the analysis of gravitational waves from compact binaries.

Paper Structure

This paper contains 31 sections, 171 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Equations of motion and action principles
  3. Action for the covariant SSC
  4. Action for a generic SSC
  5. Covariant shift of the worldline
  6. Canonical Hamiltonian
  7. Hamiltonian to linear order in spin
  8. Curvature couplings at quadratic order in spin
  9. The Kerr spacetime and its Riemann tensor
  10. Curvature couplings for a test black hole in the curvature-aligned frame
  11. Curvature couplings for general test bodies
  12. In a general frame
  13. In the time-aligned frame with the Newton-Wigner SSC
  14. Explicit canonical Hamiltonian in the time-aligned frame(s) and its post-Newtonian expansion
  15. Rotation coefficients for the spherical and Cartesian time-aligned frames
  16. ...and 16 more sections

Figures (2)

  • Figure 1: Along the worldline $z(s)$ defined by a generic SSC, with tangent $\dot z^\mu(s)$, we have the deviation vector field $\xi^\mu(s)$, which points (via the exponential map) to the worldline $\tilde{z}(s)$ defined by the covariant SSC, with tangent $\dot{\tilde{z}}^{\tilde{\mu}}(s)$.
  • Figure 2: Orders (relative to Newtonian order) of the PN-spin expansion through 4.5PN, counting $\epsilon\sim v^2\sim M/r\sim a/r\sim S/r$ (ignoring the scalings with the mass ratio) for rapidly rotating bodies.