On BMS invariance of gravitational scattering

A bstractBMS+ transformations act nontrivially on outgoing gravitational scattering data while preserving intrinsic structure at future null infinity (ℐ$$ \mathrm{\mathcal{I}} $$+). BMS− transformations similarly act on ingoing data at past null infinity (ℐ$$ \mathrm{\mathcal{I}} $$−). In this paper we apply — within a suitable finite neighborhood of the Minkowski vacuum — results of Christodoulou and Klainerman to link ℐ$$ \mathrm{\mathcal{I}} $$+ to ℐ$$ \mathrm{\mathcal{I}} $$− and thereby identify “diagonal” elements BMS0 of BMS+ × BMS−. We argue that BMS0 is a nontrivial infinite-dimensional symmetry of both classical gravitational scattering and the quantum gravity S$$ \mathcal{S} $$-matrix. It implies the conservation of net accumulated energy flux at every angle on the conformal S2 at ℐ$$ \mathrm{\mathcal{I}} $$. The associated Ward identity is shown to relate S-matrix elements with and without soft gravitons. Finally, BMS0 is recast as a U(1) Kac-Moody symmetry and an expression for the Kac-Moody current is given in terms of a certain soft graviton operator on the boundary of ℐ$$ \mathrm{\mathcal{I}} $$.