An eikonal-inspired approach to the gravitational scattering waveform

We revisit the amplitude-based derivation of gravitational waveform for the scattering of two scalar black holes at subleading post-Minkowskian (PM) order. We take an eikonal-inspired approach to the two-massive-particle cut needed in the KMOC framework, as highlighted in [1], and show that its effect is to implement a simple change of frame. This clarifies one of the points raised in [2] when comparing with the post-Newtonian (PN) results. We then provide an explicit PM expression for the waveform in the soft limit, ω → 0, including the first non-universal, ω log ω, contribution. Focusing on this regime, we show that the small-velocity limit of our result agrees with the soft limit of the PN waveform of [2], provided that the two quantities are written in the same asymptotic frame. Performing the BMS supertranslation that, as discussed in [3], is responsible for the $$\mathcal{O}$$ (G) static contribution to the asymptotic field employed in the PN literature, we find agreement between the amplitude-based and the PN soft waveform up to and including G ³/c ⁵ order.