Tidal Deformation and Dissipation of Rotating Black Holes

TL;DR

Rotating black holes do not exhibit tidal deformation as indicated by vanishing Love numbers, but they do absorb energy from external tidal fields, affecting gravitational wave signals.

Contribution

This work calculates the Love numbers of Kerr black holes, showing they vanish, and analyzes their dissipative response, clarifying boundary condition effects on gravitational wave observations.

Findings

Love numbers of Kerr black holes are zero.

Black holes exhibit non-zero dissipation due to horizon absorption.

Results are valid for all spins and multipole orders.

Abstract

We show that rotating black holes do not experience any tidal deformation when they are perturbed by a weak and adiabatic gravitational field. The tidal deformability of an object is quantified by the so-called "Love numbers", which describe the object's linear response to its external tidal field. In this work, we compute the Love numbers of Kerr black holes and find that they vanish identically. We also compute the dissipative part of the black hole's tidal response, which is non-vanishing due to the absorptive nature of the event horizon. Our results hold for arbitrary values of black hole spin, for both the electric-type and magnetic-type perturbations, and to all orders in the multipole expansion of the tidal field. The boundary conditions at the event horizon and at asymptotic infinity are incorporated in our study, as they are crucial for understanding the way in which these tidal effects are mapped onto gravitational-wave observables. In closing, we address the ambiguity issue of Love numbers in General Relativity, which we argue is resolved when those boundary conditions are taken into account. Our findings provide essential inputs for current efforts to probe the nature of compact objects through the gravitational waves emitted by binary systems.