Tidal Love Numbers of Kerr Black Holes

TL;DR

This work shows that Kerr black holes are tidally deformable under generic, slowly varying external fields, contrary to earlier static results. By solving the static Teukolsky equation for ψ0 and reconstructing the metric via a Hertz potential, the authors isolate the black hole's linear response, finding nonzero TLNs only for nonaxisymmetric perturbations and to linear order in spin. They compute explicit quadrupolar TLNs, showing λ^{M𝓔}_{2m} = λ^{S𝓑}_{2m} = i m χ (2M)^5 / 180 and relate these to the tidal torquing of spinning bodies, with no mode coupling at this order. The results are consistent with previous tidal-torque analyses and imply that spinning black holes possess small but finite tidal polarizabilities, which must be accounted for in precise gravitational-wave modeling of black-hole binaries. The work also provides a coherent, gauge-invariant framework linking curvature perturbations to Geroch-Hansen multipole moments in GR, solidifying the interpretation of black hole TLNs and their spin dependence.

Abstract

The open question of whether a Kerr black hole can become tidally deformed or not has profound implications for fundamental physics and gravitational-wave astronomy. We consider a Kerr black hole embedded in a weak and slowly varying, but otherwise arbitrary, multipolar tidal environment. By solving the static Teukolsky equation for the gauge-invariant Weyl scalar $ψ_0$, and by reconstructing the corresponding metric perturbation in an ingoing radiation gauge, for a general harmonic index $\ell$, we compute the linear response of a Kerr black hole to the tidal field. This linear response vanishes identically for a Schwarzschild black hole and for an axisymmetric perturbation of a spinning black hole. For a nonaxisymmetric perturbation of a spinning black hole, however, the linear response does not vanish, and it contributes to the Geroch-Hansen multipole moments of the perturbed Kerr geometry. As an application, we compute explicitly the rotational black hole tidal Love numbers that couple the induced quadrupole moments to the quadrupolar tidal fields, to linear order in the black hole spin, and we introduce the corresponding notion of tidal Love tensor. Finally, we show that those induced quadrupole moments are closely related to the well-known physical phenomenon of tidal torquing of a spinning body interacting with a tidal gravitational environment.