This paper studies how rotating (Kerr) black holes deform when an external tidal field varies in time. In the static limit, tidal “Love numbers” that measure how much an object bulges in response to a steady tide vanish for four‑dimensional vacuum black holes. The authors instead compute the frequency‑dependent response, called the dynamical tidal response function, which can be nonzero when the tide oscillates and when the black hole is spinning.

The team uses an effective field theory (EFT) picture in which each compact object is treated as a pointlike worldline that carries internal degrees of freedom. Those internal degrees of freedom encode finite‑size effects such as tides. To determine the EFT coupling constants, the authors solve wave‑like perturbation equations around a Kerr black hole (the full general‑relativistic problem) and match those solutions to the EFT. They focus mainly on scalar and gravitational perturbations and work to linear order in frequency — that is, to first order in the dimensionless quantity GMω (in units with G=c=1), which is small for widely separated binaries and early inspiral.

From this matching they extract the tidal response coefficients, including the “dynamical Love number” that appears at linear order in frequency for spinning black holes. The dynamical tidal response is complex: its imaginary part represents dissipation (energy lost into the hole) and its real part is the conservative correction to static Love numbers (the authors call this the conservative tidal response function). The paper gives an approximate expression for the full linear‑in‑frequency response, covering both dissipative and conservative pieces.

The authors also clarify technical issues that arise in the matching. Spin mixes different multipolar modes of the perturbations, and that mixing is essential for a consistent extraction of the response coefficients. Some intermediate steps in the calculation look singular in the case of extremal spins (the fastest possible rotation), but the authors show that the final dynamical response they obtain applies to arbitrarily large spins, including the extremal limit.