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On BMS Invariance of Gravitational Scattering

Andrew Strominger

TL;DR

This work shows that in a CK neighborhood of Minkowski space, diagonal BMS0 acts as an exact symmetry linking I− and I+ data, making the gravitational S-matrix invariant under BMS0 and enforcing angle-resolved energy conservation. It derives a Weinberg-like soft graviton Ward identity from BMS0 and recasts the symmetry as a U(1) Kac-Moody current on the celestial sphere, sourced by net radiative energy flux. The results unify asymptotic symmetry, soft theorem structure, and boundary current algebras, with implications for infrared aspects of gravity and possibly gauge theories. The approach hinges on CK stability and antipodal matching at i0 to relate in and out data and to derive concrete current-based Ward relations on the S^2 boundary of null infinity.

Abstract

BMS+ transformations act nontrivially on outgoing gravitational scattering data while preserving intrinsic structure at future null infinity (I+). BMS- transformations similarly act on ingoing data at past null infinity (I-). In this paper we apply - within a suitable finite neighborhood of the Minkowski vacuum - results of Christodoulou and Klainerman to link I+ to I- and thereby identify "diagonal" elements BMS0 of (BMS+)X(BMS-). We argue that BMS0 is a nontrivial infinite-dimensional symmetry of both classical gravitational scattering and the quantum gravity S-matrix. It implies the conservation of net accumulated energy flux at every angle on the conformal S2 at 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 null infinity.

On BMS Invariance of Gravitational Scattering

TL;DR

This work shows that in a CK neighborhood of Minkowski space, diagonal BMS0 acts as an exact symmetry linking I− and I+ data, making the gravitational S-matrix invariant under BMS0 and enforcing angle-resolved energy conservation. It derives a Weinberg-like soft graviton Ward identity from BMS0 and recasts the symmetry as a U(1) Kac-Moody current on the celestial sphere, sourced by net radiative energy flux. The results unify asymptotic symmetry, soft theorem structure, and boundary current algebras, with implications for infrared aspects of gravity and possibly gauge theories. The approach hinges on CK stability and antipodal matching at i0 to relate in and out data and to derive concrete current-based Ward relations on the S^2 boundary of null infinity.

Abstract

BMS+ transformations act nontrivially on outgoing gravitational scattering data while preserving intrinsic structure at future null infinity (I+). BMS- transformations similarly act on ingoing data at past null infinity (I-). In this paper we apply - within a suitable finite neighborhood of the Minkowski vacuum - results of Christodoulou and Klainerman to link I+ to I- and thereby identify "diagonal" elements BMS0 of (BMS+)X(BMS-). We argue that BMS0 is a nontrivial infinite-dimensional symmetry of both classical gravitational scattering and the quantum gravity S-matrix. It implies the conservation of net accumulated energy flux at every angle on the conformal S2 at 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 null infinity.

Paper Structure

This paper contains 14 sections, 79 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Vacuum-to-vacuum geometries
  3. Asymptotically flat metrics
  4. BMS transformations
  5. Christodoulou-Klainerman spaces
  6. Linking ${\mathcal{I}}^+$ to ${\mathcal{I}}^-$ near $i^0$
  7. Symmetries of classical gravitational scattering
  8. BMS invariance of the ${\cal S}$-matrix
  9. Supertranslation generators commute with ${\cal S}$
  10. Energy conservation at every angle
  11. Fock representation of BMS symmetry
  12. Supertranslations as a Kac-Moody symmetry
  13. The classical supertranslation current
  14. Kac-Moody Ward identities

Figures (1)

  • Figure 1: Penrose diagram for Minkowski space. Near ${\mathcal{I}}^+$ surfaces of constant retarded time $u$ (red) are cone-like and intersect ${\mathcal{I}}^+$ in a conformal $S^2$ parametrized by $(z,{\bar{z}})$. Cone-like surfaces of constant advanced time $v$ (green) intersect ${\mathcal{I}}^-$ in a conformal $S^2$ also parametrized by $(z,{\bar{z}})$. The future (past) $S^2$ boundary of ${\mathcal{I}}^+$ is labelled ${\mathcal{I}}^+_+$ (${\mathcal{I}}^+_-$), while the future (past) boundary of ${\mathcal{I}}^-$ is labelled ${\mathcal{I}}^-_+$ (${\mathcal{I}}^-_-$).