This paper presents a practical way to pick the asymptotic frame used to describe gravitational waves from perturbed black holes. The authors give an iterative procedure to transform a perturbative solution on a rotating (Kerr) black hole into the Bondi–Sachs gauge. That gauge uses coordinates adapted to outgoing light rays and leaves a residual freedom known as the Bondi–Metzner–Sachs (BMS) group. Fixing the BMS frame removes an important ambiguity in waveform models.

The work is aimed at calculations in black hole perturbation theory, and especially at self-force models where a small body orbits a much larger black hole. In those models the small parameter ε is the mass ratio. Until now many self-force results were in an unknown asymptotic frame and suffered from so-called infrared divergences at second perturbative order. The authors extend the Bondi–Sachs formalism to the multiscale expansions used in self-force work. They introduce the ideas of “soft hair” (low-energy gravitational degrees tied to memory) and “forgetful gauges” (gauges that hide memory), and they give a concrete iterative transformation to the Bondi–Sachs gauge that fixes the BMS frame at first order.

At a high level their approach uses the fact that the remaining gauge freedom after choosing Bondi–Sachs coordinates is exactly the BMS group of transformations of future null infinity. By prescribing a definite first-order gauge and BMS-frame choice on a Kerr background, the second-order perturbation of the radiative Weyl scalar ψ4 becomes invariant under further second-order gauge changes. Working in this first-order Bondi–Sachs gauge keeps source terms from falling off too slowly at large distances. That avoids the infrared divergences that previously forced more complicated matched far-zone expansions.

This matters because next-generation detectors such as LISA will need very accurate waveform models. Fixing the BMS frame and avoiding infrared problems lets second-order waveforms be compared consistently across different methods. The formalism also brings gravitational-wave memory into the metric in a natural way, rather than adding it by hand. The paper shows how slowly evolving supertranslations install “soft hair” on the two-body system and how that leads to the recently discussed “memory distortion,” an effect that slowly alters the emitted waves.