Gravitational self-torque and spin precession in compact binaries

TL;DR

This paper addresses how a compact body's spin precession is altered by its own gravitational field in a strong-field binary. By employing the gravitational self-force formalism and focusing on the conservative piece of the self-field, the authors compute the shift $\delta\psi$ in the geodetic precession rate to first order in the mass ratio $\mu/M$ for circular orbits around a Schwarzschild black hole. They derive a gauge-invariant expression for $\delta\psi$ and compare the results with a 3PN post-Newtonian expansion, finding good agreement in the weak-field limit but revealing significant strong-field features not captured by PN. The numerical results, obtained with two independent regularization schemes, show the expected PN structure (no $1$PN term in $\delta\psi$, agreement on 2PN and 3PN coefficients) and uncover strong-field behavior, including a sign change and a maximal $\delta\psi$ near specific radii, with implications for calibrating effective-one-body models and interpreting gravitational-wave signals from binaries.

Abstract

We calculate the effect of self-interaction on the "geodetic" spin precession of a compact body in a strong-field orbit around a black hole. Specifically, we consider the spin precession angle $ψ$ per radian of orbital revolution for a particle carrying mass $μ$ and spin $s \ll (G/c) μ^2$ in a circular orbit around a Schwarzschild black hole of mass $M \gg μ$. We compute $ψ$ through $O(μ/M)$ in perturbation theory, i.e, including the correction $δψ$ (obtained numerically) due to the torque exerted by the conservative piece of the gravitational self-field. Comparison with a post-Newtonian (PN) expression for $δψ$, derived here through 3PN order, shows good agreement but also reveals strong-field features which are not captured by the latter approximation. Our results can inform semi-analytical models of the strong-field dynamics in astrophysical binaries, important for ongoing and future gravitational-wave searches.

Figures (1)

• Figure 1: The $\mathcal{O}(\mu)$ conservative correction to $\psi$ as a function of orbital radius $r_\Omega$. The solid black line interpolates the numerical GSF data, and the dashed red and solid blue lines show the 2PN and 3PN predictions for comparison. The inset, showing the difference between the GSF and 3PN results multiplied by $(r_{\Omega}/M)^4$, hints at the value of the yet unknown 4PN coefficient. The thin brown line displays the 4PN curve assuming that the relevant coefficient is $-15/2$.