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On BMS Invariance of Gravitational Scattering
Andrew Strominger
TL;DR
Generic asymptotically flat spacetimes lack a canonical relation between BMS+ and BMS−, obstructing a direct BMS symmetry relation for the S-matrix. The paper argues that gravitational scattering is BMS-invariant and develops perturbative quantum-gravity implications. Incoming and outgoing energy fluxes match along each null generator, implying local energy conservation at every angle on the conformal sphere.
Problem
Generic asymptotically flat spacetimes lack a canonical relation between BMS+ and BMS−, obstructing a direct BMS symmetry relation for the S-matrix.
Source: §1 · pp. 2–3Method
The paper argues that gravitational scattering is BMS-invariant and develops perturbative quantum-gravity implications.
Source: §1 · p. 3Results
Incoming and outgoing energy fluxes match along each null generator, implying local energy conservation at every angle on the conformal sphere.
Source: §1 · p. 4Takeaways & Limitations
BMS0 invariance yields a soft-graviton Ward identity relating amplitudes with and without soft gravitons and a U(1) Kac-Moody formulation.
Source: §1 · p. 5, §3.2 · p. 18, §3.3 · p. 19The analysis is restricted to a finite neighborhood of Minkowski space and faces complications from classical black-hole formation and potential quantum unitarity violation.
Source: §1 · p. 3
Abstract
from arXiv · showhide
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.
1 Introduction
The paper identifies diagonal BMS0 transformations by canonically relating BMS+ and BMS− near the Minkowski vacuum, arguing that they are symmetries of classical scattering and the quantum S-matrix. Their supertranslations imply angle-by-angle energy conservation, yield soft-graviton Ward identities, and admit a U(1) Kac-Moody reformulation.
Classical and quantum symmetry
- BMS0 is argued to generate exact symmetries of classical gravitational scattering and, assuming existence, the perturbative and conjecturally nonperturbative quantum gravity S-matrix.The construction addresses the absence of a canonical BMS+–BMS− relation in generic asymptotically flat spacetimes.Source: §1 · p. 3
- Near the Minkowski vacuum, Christodoulou–Klainerman results make spatial infinity sufficiently smooth to identify BMS+ with BMS− and define diagonal generators BMS0.The paper restricts the argument to a suitably defined finite neighborhood and configurations that begin and end in the vacuum.Source: §1 · pp. 3, 5
Energy conservation
- Incoming energy flux integrated along each null generator on I− equals outgoing flux along its continuation on I+, so local energy is conserved at every angle.For interacting massless particles, soft gravitons supply localized positive or negative energy contributions that enforce this conservation.Source: §1 · p. 4
Quantum Ward identity
- The supertranslation conservation law becomes a Fock-space Ward identity relating scattering amplitudes with and without a soft graviton insertion.The paper states that this identity is precisely Weinberg’s soft graviton theorem in companion work.Source: §1 · p. 4
Kac-Moody reformulation
- The supertranslation Ward identities are recast as a U(1) Kac-Moody symmetry on the conformal sphere, with currents constructed from boundary soft gravitons.The resulting current is sourced by net accumulated radiative energy flux at each angle, and its Ward identities are equivalent to those from supertranslations.Source: §1 · pp. 4–5
2 Vacuum-to-vacuum geometries
This section focuses on weakly gravitating, vacuum-to-vacuum scattering geometries, specifying their asymptotic data, BMS freedom, and CK-space conditions. It relates the data on I− and I+ through null generators crossing i0, reducing the separate symmetries to diagonal BMS0 and generating infinite-parameter families of classical scattering solutions.
2 Vacuum-to-vacuum geometries
- The analysis considers weakly gravitating geometries that begin and end in the vacuum, with asymptotic flatness imposed at both I+ and I−.The section establishes conventions for the spaces under consideration.Source: §2 · p. 5, §2.1 · p. 6
2.1 Asymptotically flat metrics
- Asymptotically flat metrics are described in retarded and advanced Bondi coordinates, with Bondi mass aspect mB encoding local energy and Bondi news Nzz encoding outgoing radiation.I+ and I− are null boundaries whose generators meet conformal S2 cross-sections, while their relation near spatial infinity is generally subtle.Source: §2.1 · pp. 5, 6, 8
2.4 Linking I+ to I−near i0
- Near i0, gravitational scattering requires relating incoming I− data to outgoing I+ data despite a subtle conformal singularity and BMS+×BMS− ambiguity.The leading I+ fields mB and Czz are determined by integrating null-infinity constraint equations from suitable initial conditions.Source: §2.1 · p. 8, §2.4 · p. 12
2.2 BMS transformations
- BMS transformations comprise six asymptotic SL(2,C) generators and infinitely many commuting supertranslations, but the connection between BMS+ and BMS− remains unspecified here.Global conformal transformations and supertranslations act through the asymptotic metric data on null infinity.Source: §2.2 · pp. 8–9
2.3 Christodoulou-Klainerman spaces
- CK spaces describe globally geodesically complete, asymptotically flat solutions reverting to vacuum, obtained under a global smallness condition below black-hole formation.The smallness condition holds in a finite neighborhood of Minkowski space, establishing Minkowski stability.Source: §2.3 · pp. 9–10
2.4 Linking I+ to I−near i0
- Linking data along null generators through i0 imposes a condition that breaks the separate BMS symmetries of I+ and I− to a single diagonal symmetry.The identification labels points on I+ and I− by the same (z, z̄) coordinate, with an antipodal relation between the boundaries.Source: §2.4 · pp. 13–14
- The construction requires CK-space properties: if the news tensor falls off as |u|^-1, Czz diverges logarithmically and the key boundary condition cannot be imposed.Thus the linking of I+ and I− may fail for more general asymptotically flat spaces.Source: §2.4 · p. 14
2.5 Symmetries of classical gravitational scattering
- Diagonal SL(2,C) transformations boost or rotate both initial and final data, while the four special supertranslations translate the initial and final data consistently.Generic supertranslations can also alter the relative timing of ingoing and outgoing wave pulses while preserving the scattering relation.Source: §2.5 · p. 15
- For every CK space, diagonal BMS0 generates an infinite-parameter family of scattering solutions, differing through ordinary Poincare transformations and less-obvious supertranslations.The construction starts by integrating constraint equations on I+ and I− to obtain final data from initial data, then applies diagonal BMS transformations.Source: §2.5 · pp. 14–15
3 BMS invariance of the S-matrix
The paper argues that diagonal BMS0 transformations commute with the gravitational S-matrix and derive angle-by-angle energy conservation and soft-graviton Ward identities. It then represents these relations semiclassically in a Fock basis, while noting assumptions about applying the derivation to Fock-space scattering.
3.1 Supertranslation generators commute with S
- The supertranslation action combines transformations of radiative data with the addition of a soft graviton.The soft mode is identified with the zero-frequency limit of a metric perturbation and is associated with non-constant supertranslations acting nontrivially on the vacuum.Source: §3.1 · pp. 16–17
- Infinitesimal BMS0 transformations commute with the S-matrix, establishing its infinitesimal BMS0 invariance.The derivation uses supertranslation generators on I+ and I− and the relation between them on weakly gravitating CK spaces.Source: §3 · p. 15, §3.1 · p. 17
3.2 Energy conservation at every angle
- At every angle on the conformal S2, incoming accumulated energy equals outgoing accumulated energy.This conservation law depends on the linear soft-graviton contribution to the unintegrated Bondi energy, which is a total sphere derivative and does not affect total integrated energy.Source: §3.2 · p. 18
3.1 Supertranslation generators commute with S
- The analysis is restricted to weakly gravitating CK spaces and omits a late-time mass-aspect term that may matter in other applications.The omitted term vanishes for the classical solutions and semiclassical states considered here.Source: §3.1 · p. 16, §3.3 · p. 18
3.3 Fock representation of BMS symmetry
- The Fock-space application is assumed rather than derived directly because perturbative scattering configurations differ from the field configurations used in the classical derivation.The resulting expressions are supported by their equivalence to Weinberg’s soft graviton theorem [8].Source: §3.3 · p. 18
- In the Fock representation, BMS invariance yields a Ward identity relating S-matrix elements with and without a soft graviton insertion.The relation is identified with Weinberg’s soft graviton theorem in the companion paper.Source: §3.3 · p. 19
4 Supertranslations as a Kac-Moody symmetry
The paper recasts supertranslations as a non-compact U(1) Kac-Moody symmetry on the asymptotic S2. A classical current is sourced by local outgoing-minus-incoming flux, and its quantum Ward identities encode the supertranslation S-matrix relations.
4 Supertranslations as a Kac-Moody symmetry
- Supertranslation generators can be recast as those of a non-compact Kac-Moody algebra acting on the asymptotic S2 at null infinity.This construction is presented as compatible with proposals for a Virasoro action on the same S2.Source: §4 · p. 20
4.1 The classical supertranslation current
- The classical current is locally sourced by the difference between outgoing and incoming accumulated radiation at a point on the conformal S2.Its classical Ward identity is identified with the Ward identity of a U(1) Kac-Moody current.Source: §4.1 · p. 20
4.2 Kac-Moody Ward identities
- The semiclassical current Pz has charge given by the net accumulated energy flux across a point on the conformal S2.Its insertion obeys the same supertranslation Ward identity as a U(1) Kac-Moody current.Source: §4.2 · p. 21
- Contour insertions of Pz generate a noncompact U(1) supertranslation Kac-Moody algebra and capture the S-matrix information of supertranslation invariance.The information is repackaged rather than represented by Pz as the supertranslation generator on the incoming or outgoing Hilbert spaces.Source: §4.2 · pp. 21–22
- Negative-helicity soft-graviton insertions do not provide new S-matrix information because their difference from positive-helicity insertions vanishes for real supertranslation profiles.This motivates focusing on the holomorphic current Pz, while the antiholomorphic current may be redundant.Source: §4.2 · p. 22
- The asymptotic symmetry is therefore suggested to be a single U(1) Kac-Moody symmetry on the conformal S2.The conclusion concerns the S-matrix realization and does not rule out other uses of the antiholomorphic current.Source: §4.2 · p. 22