Tidal invariants for compact binaries on quasi-circular orbits

TL;DR

This paper extends the gravitational self-force method to include tidal effects for compact bodies in quasi-circular orbits around black holes, providing gauge-invariant tidal quantities and numerical results in strong-field regimes.

Contribution

It introduces a gauge-invariant framework for tidal effects at first order in mass ratio and computes numerical results for Kerr and Schwarzschild spacetimes, enhancing gravitational wave modeling.

Findings

Consistent with post-Newtonian expansions in weak fields

Identifies additional structure in strong-field regimes

Provides numerical data up to the light-ring radius

Abstract

We extend the gravitational self-force approach to encompass `self-interaction' tidal effects for a compact body of mass $\mu$ on a quasi-circular orbit around a black hole of mass $M \gg \mu$. Specifically, we define and calculate at $O(\mu)$ (conservative) shifts in the eigenvalues of the electric- and magnetic-type tidal tensors, and a (dissipative) shift in a scalar product between their eigenbases. This approach yields four gauge-invariant functions, from which one may construct other tidal quantities such as the curvature scalars and the speciality index. First, we analyze the general case of a geodesic in a regular perturbed vacuum spacetime admitting a helical Killing vector and a reflection symmetry. Next, we specialize to focus on circular orbits in the equatorial plane of Kerr spacetime at $O(\mu)$. We present accurate numerical results for the Schwarzschild case for orbital radii up to the light-ring, calculated via independent implementations in Lorenz and Regge-Wheeler gauges. We show that our results are consistent with leading-order post-Newtonian expansions, and demonstrate the existence of additional structure in the strong-field regime. We anticipate that our strong-field results will inform (e.g.) effective one-body models for the gravitational two-body problem that are invaluable in the ongoing search for gravitational waves.