Response of a Kerr black hole to a generic tidal perturbation

TL;DR

The paper advances our understanding of how Kerr black holes respond to external tidal fields across scalar, electromagnetic, and gravitational perturbations by solving the Teukolsky equation in the near-zone and small-frequency regime and employing analytic continuation in the angular quantum number. The authors show that static tidal Love numbers vanish for Kerr, and that dynamical TLNs vanish in axisymmetric or Schwarzschild-like limits, while non-axisymmetric perturbations generally yield non-zero dynamical TLNs for both non-extremal and extremal Kerr, with logarithmic contributions arising in certain non-extremal EM and gravitational cases. They further demonstrate that TLNs depend on the spin weight $s$ and can change when $s\to -s$, and treat extremal Kerr separately, where non-zero dynamical TLNs persist for non-axisymmetric perturbations but axisymmetric cases still vanish. These results have potential implications for gravitational-wave astrophysics and tests of General Relativity in strong-field regimes, offering a precise characterization of how black holes deform under tidal forces. The work also clarifies the distinction between static and dynamic tidal responses and connects relativistic perturbation theory with EFT approaches via near-zone matching.

Abstract

We derive the response, the real part of which provides the tidal Love numbers, for non-extremal as well as extremal Kerr black holes under generic tidal perturbations. Our results suggest that the static as well as dynamical (linear-in-frequency) Love numbers vanish for both Schwarzschild and slowly rotating (linear-in-angular momentum) Kerr black holes, under generic perturbations. The vanishing of static and dynamical Love numbers also holds for axisymmetric tidal perturbations of non-extremal and extremal Kerr black holes. In fact, even under generic tidal perturbations, the static Love numbers of Kerr black holes vanish identically. The only case with non-zero Love numbers corresponds to the non-axisymmetric dynamical tidal perturbations of Kerr black holes (to arbitrary order of angular momentum). We also demonstrate that the non-zero dynamical Love numbers, for both non-extremal and extremal Kerr black holes, get modified under the change of the sign of the spin-weight, for electromagnetic and gravitational tidal perturbations.

Figures (2)

• Figure 1: Contour plots of TLNs (with non-logarithmic term) associated with a non-extremal Kerr BH have been presented with dimensionless frequency $M\omega$ and dimensionless rotation parameter $(a/M)$, for EM and gravitational perturbations with different spin-weights $s$. In the case of negative spin weights, for both EM and gravitational perturbations, TLNs can be zero, positive, or negative for some particular values of the frequency and rotation parameter. The zero TLN contours are shown with dashed lines in \ref{['fig:1a']} and \ref{['fig:1c']}. Note that for EM perturbation we have taken $l=1=m$, while for gravitational perturbation, we have $l=2=m$. We would also like to point out that our analysis is only valid for small mode frequencies, satisfying $M\omega \ll 1$. Keeping this in mind, we have presented the TLNs in the above plots till $M\omega\sim 0.1$. However, even for $M\omega \sim \mathcal{O}(0.1)$, the mode frequency is small but may not still be considered as a very small quantity. Thus, the features present here may also be influenced by non-linear terms, which we have ignored in our analysis.
• Figure 2: Contour plots of the TLNs $\,_{0}k_{11}$ (associated with the $l=1=m$ mode of the scalar perturbation) and $\,_{0}k_{22}$ (associated with the $l=2=m$ mode of the scalar perturbation) are presented against the dimensionless frequency $M\omega$ and the dimensionless rotation parameter $(a/M)$. As is evident, the TLNs are negative, and their magnitudes increase with increasing rotation. Here, too, we have plotted the TLNs for mode frequencies up to $M\omega\sim 0.1$, which, though small, may not be considered as very small compared to unity. Thus, the behavior of the TLNs at the rightmost region in these plots may be affected by non-linear terms we have ignored in our analysis.