Classical Soft Graviton Theorem Rewritten

TL;DR

The paper proves that in classical scattering, the coefficients of the $1/u$ and $\ln u/u^2$ terms in the late-time gravitational waveform are determined solely by the incoming momenta, remaining independent of final-state massless radiation even when black-hole formation occurs. It extends the known cancellation for the $1/u$ term to the subsubleading $\ln u/u^2$ term by a parallel rewriting that partitions the final-state momenta into massive and massless sectors and demonstrates cancellations among contributions. An explicit massless scattering example confirms the predicted relations among the coefficients and their proper normalization for low-frequency flux. The work also provides a detailed angular-integrated, low-frequency energy flux calculation for large-impact-parameter massless scattering, illustrating the practical implications for gravitational radiation predictions in classical high-energy processes.

Abstract

Classical soft graviton theorem gives the gravitational wave-form at future null infinity at late retarded time $u$ for a general classical scattering. The large $u$ expansion has three known universal terms: the constant term, the term proportional to $1/u$ and the term proportional to $\ln u/u^2$, whose coefficients are determined solely in terms of the momenta of incoming and the outgoing hard particles, including the momenta carried by outgoing gravitational and electromagnetic radiation produced during scattering. For the constant term, also known as the memory effect, the dependence on the momenta carried away by the final state radiation / massless particles is known as non-linear memory or null memory. It was shown earlier that for the coefficient of the $1/u$ term the dependence on the momenta of the final state massless particles / radiation cancels and the result can be written solely in terms of the momenta of the incoming particles / radiation and the final state massive particles. In this note we show that the same result holds for the coefficient of the $\ln u/u^2$ term. Our result implies that for scattering of massless particles the coefficients of the $1/u$ and $\ln u/u^2$ terms are determined solely by the incoming momenta, even if the particles coalesce to form a black hole and massless radiation. We use our result to compute the low frequency flux of gravitational radiation from the collision of massless particles at large impact parameter.