On BMS invariance of gravitational scattering

Abstract

BMS+ transformations act nontrivially on outgoing gravitational scattering data while preserving intrinsic structure at future null infinity ( $$ \mathrm{\mathcal{I}} $$ +). BMS− transformations similarly act on ingoing data at past null infinity ( $$ \mathrm{\mathcal{I}} $$ −). In this paper we apply — within a suitable finite neighborhood of the Minkowski vacuum — results of Christodoulou and Klainerman to link $$ \mathrm{\mathcal{I}} $$ + to $$ \mathrm{\mathcal{I}} $$ − 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 $$ \mathcal{S} $$ -matrix. It implies the conservation of net accumulated energy flux at every angle on the conformal S 2 at $$ \mathrm{\mathcal{I}} $$ . 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 $$ \mathrm{\mathcal{I}} $$ .