Dynamical tidal Love numbers of black holes under generic perturbations: Connecting black hole perturbation theory with effective field theory

TL;DR

This paper develops an effective field theory approach to analyze the dynamical tidal response of spinning black holes, deriving the frequency-dependent Love numbers and connecting perturbation theory with EFT methods.

Contribution

It introduces a systematic EFT framework for modeling black hole tidal responses, including spin effects, and matches it with full perturbation solutions to determine Love numbers.

Findings

Derived the dynamical Love number for spinning black holes.

Provided an approximate expression for the frequency-dependent tidal response.

Identified key subtleties in mode mixing due to black hole spin.

Abstract

The foundation for modeling the coupling of the internal structure of compact objects in binary systems to their dynamics and emitted gravitational waves is a systematic effective field theory (EFT) framework, where each compact object is replaced by a worldline endowed with a set of internal degrees of freedom. These degrees of freedom encode finite-size effects and thereby distinguish between different classes of compact objects. Among finite-size effects, tidal interactions play a central role, as they are associated to various kinds of deformations of a body under the influence of external tidal fields. In this work, we analyze the dynamical tidal response of Kerr black holes to generic-spin perturbations, focusing primarily on the scalar and gravitational cases, and working to linear order in frequency. We establish an EFT description of the perturbed black hole that accounts for the couplings between the spin, gravitoelectric and -magnetic tidal fields. We match this to wave-like solutions to the full black hole perturbation equations in order to determine the tidal response coefficients. In particular, we obtain the dynamical Love number, which appears at linear order in frequency for spinning black holes, and derive an approximate expression for the dynamical tidal response, including both dissipative and conservative pieces. We also examine several technical subtleties that arise in the matching procedure, with special emphasis on the mixing of multipolar modes induced by the spin of the compact object, which proves to be essential for a consistent treatment.