Tidal Deformation and Dissipation of Rotating Black Holes

TL;DR

The work analyzes how rotating black holes respond to external tidal fields by examining the $\psi_4$ Weyl scalar and solving the Teukolsky equation within a near/far-region framework under horizon and infinity boundary conditions. It shows that Kerr black holes have vanishing Love numbers for all spins and perturbation types, while their tidal dissipation is nonzero, given by $F^{I, \mathrm{Kerr}}_{\ell m}= - i P_+ \frac{(\ell-2)!(\ell+2)!}{(2\ell)!(2\ell+1)!} \prod_{j=1}^\ell \left[ j^2 + 4 P_+^2 \right]$ with $P_+ = (a m - 2 r_+ M \omega)/(r_+ - r_-)$. This boundary-condition–consistent analysis resolves prior ambiguities in Love numbers and connects tidal effects to gravitational-wave observables through fluxes at the horizon and at infinity, providing robust inputs for GW astrophysics and tests of black-hole nature.

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.

Figures (1)

• Figure 1: Illustration of the different regions of $\psi_4$ for a rotating black hole perturbed by an external tidal field. The near region encodes the purely-ingoing boundary condition at the event horizon $r=r_+$, while the far region captures the outgoing waves at asymptotic infinity $r \to \infty$. The tidal perturbation is sourced in the intermediate region, which captures neither of those conditions. The widths of the regions are determined by two lengthscales: i): the radius of curvature of the tidal environment, $\mathcal{R}$, beyond which the multipole expansion of the tidal field breaks down, and ii) the radius $N r_+$, where $N>1$ is a constant value, suitably chosen such that the Kerr background is well described by the Schwarzschild metric at larger distances.