BMS supertranslations and Weinberg’s soft graviton theorem

  • Temple He
  • Vyacheslav Lysov
  • Prahar MitraEmail author
  • Andrew Strominger
Open Access
Regular Article - Theoretical Physics


Recently it was conjectured that a certain infinite-dimensional “diagonal” subgroup of BMS supertranslations acting on past and future null infinity ( Open image in new window and Open image in new window ) is an exact symmetry of the quantum gravity S-matrix, and an associated Ward identity was derived. In this paper we show that this supertranslation Ward identity is precisely equivalent to Weinberg’s soft graviton theorem. Along the way we construct the canonical generators of supertranslations at Open image in new window , including the relevant soft graviton contributions. Boundary conditions at the past and future of Open image in new window and a correspondingly modified Dirac bracket are required. The soft gravitons enter as boundary modes and are manifestly the Goldstone bosons of spontaneously broken supertranslation invariance.


Scattering Amplitudes Space-Time Symmetries 


Open Access

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  1. [1]
    S. Weinberg, Infrared photons and gravitons, Phys. Rev. 140 (1965) B516.ADSMathSciNetCrossRefGoogle Scholar
  2. [2]
    S. Weinberg, The quantum theory of fields. Volume 1: foundations, Cambridge University Press, Cambridge U.K. (1995).CrossRefGoogle Scholar
  3. [3]
    A. Strominger, On BMS invariance of gravitational scattering, JHEP 07 (2014) 152 [arXiv:1312.2229] [INSPIRE].ADSCrossRefGoogle Scholar
  4. [4]
    H. Bondi, M.G.J. van der Burg and A.W.K. Metzner, Gravitational waves in general relativity. 7. Waves from axisymmetric isolated systems, Proc. Roy. Soc. Lond. A 269 (1962) 21 [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  5. [5]
    R.K. Sachs, Gravitational waves in general relativity. 8. Waves in asymptotically flat space-times, Proc. Roy. Soc. Lond. A 270 (1962) 103 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    A. Strominger, Asymptotic symmetries of Yang-Mills theory, JHEP 07 (2014) 151 [arXiv:1308.0589] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    J. Maldacena and A. Zhiboedov, Notes on soft factors, unpublished (2012).Google Scholar
  8. [8]
    J. Maldacena and A. Zhiboedov, private communication.Google Scholar
  9. [9]
    D. Christodoulou and S. Klainerman, The global nonlinear stability of the Minkowski space, Princeton University Press, Princeton U.S.A. (1993).zbMATHGoogle Scholar
  10. [10]
    G. Barnich and C. Troessaert, Symmetries of asymptotically flat 4 dimensional spacetimes at null infinity revisited, Phys. Rev. Lett. 105 (2010) 111103 [arXiv:0909.2617] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  11. [11]
    G. Barnich and C. Troessaert, Supertranslations call for superrotations, PoS 21 (2010) 010 [arXiv:1102.4632] [INSPIRE].Google Scholar
  12. [12]
    G. Barnich and C. Troessaert, BMS charge algebra, JHEP 12 (2011) 105 [arXiv:1106.0213] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    G. Barnich and C. Troessaert, Comments on holographic current algebras and asymptotically flat four dimensional spacetimes at null infinity, JHEP 11 (2013) 003 [arXiv:1309.0794] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    T. Banks, A critique of pure string theory: heterodox opinions of diverse dimensions, hep-th/0306074 [INSPIRE].
  15. [15]
    A. Ashtekar and R.O. Hansen, A unified treatment of null and spatial infinity in general relativity. IUniversal structure, asymptotic symmetries and conserved quantities at spatial infinity, J. Math. Phys. 19 (1978) 1542 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  16. [16]
    A. Ashtekar, Asymptotic quantization of the gravitational field, Phys. Rev. Lett. 46 (1981) 573 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  17. [17]
    A. Ashtekar and M. Streubel, Symplectic geometry of radiative modes and conserved quantities at null infinity, Proc. Roy. Soc. Lond. A 376 (1981) 585 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  18. [18]
    A. Ashtekar, Asymptotic quantization: based on 1984 Naples lectures, Bibliopolis, Naples Italy (1987).zbMATHGoogle Scholar
  19. [19]
    S.Y. Choi, J.S. Shim and H.S. Song, Factorization and polarization in linearized gravity, Phys. Rev. D 51 (1995) 6 [hep-th/9411092] [INSPIRE].Google Scholar

Copyright information

© The Author(s) 2015

Authors and Affiliations

  • Temple He
    • 1
  • Vyacheslav Lysov
    • 1
  • Prahar Mitra
    • 1
    Email author
  • Andrew Strominger
    • 1
  1. 1.Center for the Fundamental Laws of NatureHarvard UniversityCambridgeU.S.A.

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