[Paper Review] Precisely computing bound orbits of spinning bodies around black holes I: General framework and results for nearly equatorial orbits

Lisa V. Drummond, Scott A. Hughes|arXiv (Cornell University)|Jan 31, 2022

Pulsars and Gravitational Waves ResearchPhysics and Astronomy143 references41 citations

TL;DR

This paper develops a frequency-domain method to precisely compute bound orbits of spinning bodies around Kerr black holes, incorporating spin-curvature forces to linear order in spin. It demonstrates that spin induces measurable shifts in orbital frequencies Ωr and Ωφ, which directly affect gravitational wave phasing—critical for accurate EMRI waveform modeling in LISA observations.

ABSTRACT

Very large mass ratio binary black hole systems are of interest both as a clean limit of the two-body problem in general relativity, as well as for their importance as sources of low-frequency gravitational waves. At lowest order, the smaller body moves along a geodesic of the larger black hole's spacetime. Post-geodesic effects include the gravitational self force, which incorporates the backreaction of gravitational-wave emission, and the spin-curvature force, which arises from coupling of the small body's spin to the black hole's spacetime curvature. In this paper, we describe a method for precisely computing bound orbits of spinning bodies about black holes. Our analysis builds off of pioneering work by Witzany which demonstrated how to describe the motion of a spinning body to linear order in the small body's spin. Exploiting the fact that in the large mass-ratio limit spinning-body orbits are close to geodesics and using closed-form results due to van de Meent describing precession of the small body's spin along black hole orbits, we develop a frequency-domain formulation of the motion which can be solved very precisely. We examine a range of orbits with this formulation, focusing in this paper on orbits which are eccentric and nearly equatorial (i.e., the orbit's motion is $\mathcal{O}(S)$ out of the equatorial plane), but for which the small body's spin is arbitrarily oriented. We discuss generic orbits with general small-body spin orientation in a companion paper. We characterize the behavior of these orbits and show how the small body's spin shifts the frequencies $\Omega_r$ and $\Omega_\phi$ which affect orbital motion. These frequency shifts change accumulated phases which are direct gravitational-wave observables, illustrating the importance of precisely characterizing these quantities for gravitational-wave observations. (Abridged)

Motivation & Objective

• To develop a high-precision computational framework for bound orbits of spinning bodies in the extreme mass-ratio limit.
• To incorporate spin-curvature forces in the motion of small bodies orbiting Kerr black holes, valid to linear order in the small body's spin.
• To characterize how spin alters orbital frequencies Ωr and Ωφ, which are key observables in gravitational wave signals.
• To enable accurate waveform modeling for extreme mass-ratio inspirals (EMRIs) detectable by LISA.
• To lay the foundation for modeling generic spinning-body orbits with arbitrary spin orientation in a companion paper.

Proposed method

• Uses a frequency-domain formulation to solve the equations of motion for spinning bodies in Kerr spacetime.
• Exploits the fact that spinning-body orbits are close to geodesics in the large mass-ratio limit.
• Applies closed-form results from van de Meent for spin precession along geodesics to model spin evolution.
• Solves the Mathisson-Papapetrou equations with the spin-curvature force to linear order in spin.
• Focuses on nearly equatorial, eccentric orbits with arbitrary spin orientation.
• Computes orbital frequency shifts Ωr and Ωφ due to spin, which affect accumulated gravitational wave phases.

Experimental results

Research questions

• RQ1 How do spin-curvature forces modify the orbital frequencies Ωr and Ωφ of a spinning body in a nearly equatorial orbit around a Kerr black hole?
• RQ2 What is the magnitude and structure of spin-induced corrections to orbital motion in the extreme mass-ratio limit?
• RQ3 How do arbitrary spin orientations affect the precession and frequency shifts of bound orbits compared to geodesic motion?
• RQ4 To what extent can frequency-domain methods achieve high-precision computation of spinning-body orbits?
• RQ5 How do these spin-induced frequency shifts impact the gravitational wave phasing relevant for LISA observations?

Key findings

• Spin-curvature forces induce measurable shifts in the radial frequency Ωr and azimuthal frequency Ωφ of bound orbits, even for small spin.
• These frequency shifts are directly observable in gravitational wave signals and affect accumulated phases over thousands of orbits.
• The frequency shifts depend on the orientation of the small body's spin relative to the orbital plane and black hole spin.
• The method achieves high-precision computation of orbits by leveraging the smallness of the mass ratio and known spin precession solutions.
• The framework enables accurate modeling of orbital dynamics beyond geodesics, essential for EMRI template development.
• The results show that spin effects are non-negligible for LISA-band EMRI observations, even at linear order in spin.