Journal of High Energy Physics

, 2018:200 | Cite as

Superboost transitions, refraction memory and super-Lorentz charge algebra

  • Geoffrey Compère
  • Adrien Fiorucci
  • Romain Ruzziconi
Open Access
Regular Article - Theoretical Physics


We derive a closed-form expression of the orbit of Minkowski spacetime under arbitrary Diff(S2) super-Lorentz transformations and supertranslations. Such vacua are labelled by the superboost, superrotation and supertranslation fields. Impulsive transitions among vacua are related to the refraction memory effect and the displacement memory effect. A phase space is defined whose asymptotic symmetry group consists of arbitrary Diff(S2) super-Lorentz transformations and supertranslations. It requires a renormalization of the symplectic structure. We show that our final surface charge expressions are consistent with the leading and subleading soft graviton theorems. We contrast the leading BMS triangle structure to the mixed overleading/subleading BMS square structure.


Classical Theories of Gravity Gauge Symmetry Space-Time Symmetries 


Open Access

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  1. [1]
    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].
  2. [2]
    R.K. Sachs, Gravitational waves in general relativity. 8. Waves in asymptotically flat space-times, Proc. Roy. Soc. Lond. A 270 (1962) 103 [INSPIRE].
  3. [3]
    T. He, V. Lysov, P. Mitra and A. Strominger, BMS supertranslations and Weinberg’s soft graviton theorem, JHEP 05 (2015) 151 [arXiv:1401.7026] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    A. Strominger, On BMS Invariance of Gravitational Scattering, JHEP 07 (2014) 152 [arXiv:1312.2229] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  5. [5]
    A. Strominger and A. Zhiboedov, Gravitational Memory, BMS Supertranslations and Soft Theorems, JHEP 01 (2016) 086 [arXiv:1411.5745] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    S. Weinberg, Infrared photons and gravitons, Phys. Rev. 140 (1965) B516 [INSPIRE]. [7] Y.B. Zel’dovich and A.G. Polnarev, Radiation of gravitational waves by a cluster of superdense stars, Sov. Astron. 18 (1974) 17.
  7. [7]
    D. Christodoulou, Nonlinear nature of gravitation and gravitational wave experiments, Phys. Rev. Lett. 67 (1991) 1486 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    L. Blanchet and T. Damour, Tail Transported Temporal Correlations in the Dynamics of a Gravitating System, Phys. Rev. D 37 (1988) 1410 [INSPIRE].
  9. [9]
    L. Blanchet and T. Damour, Hereditary effects in gravitational radiation, Phys. Rev. D 46 (1992) 4304 [INSPIRE].
  10. [10]
    J. Frauendiener, Note on the memory effect, Class. Quant. Grav. 9 (1992) 1639.Google Scholar
  11. [11]
    A. Strominger, Lectures on the Infrared Structure of Gravity and Gauge Theory, arXiv:1703.05448 [INSPIRE].
  12. [12]
    G. Compère and A. Fiorucci, Advanced Lectures in General Relativity, arXiv:1801.07064 [INSPIRE].
  13. [13]
    F. Cachazo and A. Strominger, Evidence for a New Soft Graviton Theorem, arXiv:1404.4091 [INSPIRE].
  14. [14]
    D.A. Nichols, Center-of-mass angular momentum and memory effect in asymptotically flat spacetimes, Phys. Rev. D 98 (2018) 064032 [arXiv:1807.08767] [INSPIRE].
  15. [15]
    S. Pasterski, A. Strominger and A. Zhiboedov, New Gravitational Memories, JHEP 12 (2016) 053 [arXiv:1502.06120] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    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
  17. [17]
    M. Campiglia and A. Laddha, Asymptotic symmetries and subleading soft graviton theorem, Phys. Rev. D 90 (2014) 124028 [arXiv:1408.2228] [INSPIRE].
  18. [18]
    J. de Boer and S.N. Solodukhin, A Holographic reduction of Minkowski space-time, Nucl. Phys. B 665 (2003) 545 [hep-th/0303006] [INSPIRE].
  19. [19]
    G. Barnich and C. Troessaert, Aspects of the BMS/CFT correspondence, JHEP 05 (2010) 062 [arXiv:1001.1541] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    G. Barnich and C. Troessaert, BMS charge algebra, JHEP 12 (2011) 105 [arXiv:1106.0213] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    M. Campiglia and A. Laddha, New symmetries for the Gravitational S-matrix, JHEP 04 (2015) 076 [arXiv:1502.02318] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    R.M. Wald and A. Zoupas, A General definition of ‘conserved quantities’ in general relativity and other theories of gravity, Phys. Rev. D 61 (2000) 084027 [gr-qc/9911095] [INSPIRE].
  23. [23]
    T. Regge and C. Teitelboim, Role of Surface Integrals in the Hamiltonian Formulation of General Relativity, Annals Phys. 88 (1974) 286 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    V. Iyer and R.M. Wald, Some properties of Noether charge and a proposal for dynamical black hole entropy, Phys. Rev. D 50 (1994) 846 [gr-qc/9403028] [INSPIRE].
  25. [25]
    G. Barnich and F. Brandt, Covariant theory of asymptotic symmetries, conservation laws and central charges, Nucl. Phys. B 633 (2002) 3 [hep-th/0111246] [INSPIRE].
  26. [26]
    G. Compere and D. Marolf, Setting the boundary free in AdS/CFT, Class. Quant. Grav. 25 (2008) 195014 [arXiv:0805.1902] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  27. [27]
    D. Kapec, V. Lysov, S. Pasterski and A. Strominger, Semiclassical Virasoro symmetry of the quantum gravity S-matrix, JHEP 08 (2014) 058 [arXiv:1406.3312] [INSPIRE].
  28. [28]
    E. Conde and P. Mao, BMS Supertranslations and Not So Soft Gravitons, JHEP 05 (2017) 060 [arXiv:1612.08294] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  29. [29]
    S.W. Hawking, M.J. Perry and A. Strominger, Superrotation Charge and Supertranslation Hair on Black Holes, JHEP 05 (2017) 161 [arXiv:1611.09175] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  30. [30]
    L. Blanchet and T. Damour, Radiative gravitational fields in general relativity I. general structure of the field outside the source, Phil. Trans. Roy. Soc. Lond. A 320 (1986) 379 [INSPIRE].
  31. [31]
    H. Stephani, D. Kramer, M. MacCallum, C. Hoenselaers and E. Herlt, Exact solutions of Einstein’s field equations, Cambridge University Press (2003) [INSPIRE].
  32. [32]
    R. Penrose, The geometry of impulsive gravitational waves, in General Relativity, Papers in Honour of J.L. Synge, Clarendon Press (1972), pg. 101.Google Scholar
  33. [33]
    Y. Nutku and R. Penrose, On Impulsive Gravitational Waves, Twistor Newslett. 34 (1992) 9.Google Scholar
  34. [34]
    J. Podolsky and J.B. Griffiths, Expanding impulsive gravitational waves, Class. Quant. Grav. 16 (1999) 2937 [gr-qc/9907022] [INSPIRE].
  35. [35]
    J.B. Griffiths, J. Podolsky and P. Docherty, An Interpretation of Robinson-Trautman type N solutions, Class. Quant. Grav. 19 (2002) 4649 [gr-qc/0208022] [INSPIRE].
  36. [36]
    J.B. Griffiths and P. Docherty, A Disintegrating cosmic string, Class. Quant. Grav. 19 (2002) L109 [gr-qc/0204085] [INSPIRE].
  37. [37]
    A. Strominger and A. Zhiboedov, Superrotations and Black Hole Pair Creation, Class. Quant. Grav. 34 (2017) 064002 [arXiv:1610.00639] [INSPIRE].
  38. [38]
    G. Compère and J. Long, Vacua of the gravitational field, JHEP 07 (2016) 137 [arXiv:1601.04958] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  39. [39]
    É. É. Flanagan and D.A. Nichols, Conserved charges of the extended Bondi-Metzner-Sachs algebra, Phys. Rev. D 95 (2017) 044002 [arXiv:1510.03386] [INSPIRE].
  40. [40]
    G. Compère and J. Long, Classical static final state of collapse with supertranslation memory, Class. Quant. Grav. 33 (2016) 195001 [arXiv:1602.05197] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  41. [41]
    J. Podolsky and R. Steinbauer, Geodesics in space-times with expanding impulsive gravitational waves, Phys. Rev. D 67 (2003) 064013 [gr-qc/0210007] [INSPIRE].
  42. [42]
    J. Podolsky and R. Svarc, Refraction of geodesics by impulsive spherical gravitational waves in constant-curvature spacetimes with a cosmological constant, Phys. Rev. D 81 (2010) 124035 [arXiv:1005.4613] [INSPIRE].
  43. [43]
    J. Podolsky, C. Sämann, R. Steinbauer and R. Svarc, The global uniqueness and C 1 -regularity of geodesics in expanding impulsive gravitational waves, Class. Quant. Grav. 33 (2016) 195010 [arXiv:1602.05020] [INSPIRE].
  44. [44]
    H. Bondi, Plane gravitational waves in general relativity, Nature 179 (1957) 1072 [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  45. [45]
    P.M. Zhang, C. Duval, G.W. Gibbons and P.A. Horvathy, The Memory Effect for Plane Gravitational Waves, Phys. Lett. B 772 (2017) 743 [arXiv:1704.05997] [INSPIRE].
  46. [46]
    P.M. Zhang, C. Duval, G.W. Gibbons and P.A. Horvathy, Soft gravitons and the memory effect for plane gravitational waves, Phys. Rev. D 96 (2017) 064013 [arXiv:1705.01378] [INSPIRE].
  47. [47]
    P.M. Zhang, C. Duval, G.W. Gibbons and P.A. Horvathy, Velocity Memory Effect for Polarized Gravitational Waves, JCAP 05 (2018) 030 [arXiv:1802.09061] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  48. [48]
    P.M. Zhang, M. Elbistan, G.W. Gibbons and P.A. Horvathy, Sturm-Liouville and Carroll: at the heart of the memory effect, Gen. Rel. Grav. 50 (2018) 107 [arXiv:1803.09640] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  49. [49]
    J. Distler, R. Flauger and B. Horn, Double-soft graviton amplitudes and the extended BMS charge algebra, arXiv:1808.09965 [INSPIRE].
  50. [50]
    D. Christodoulou and S. Klainerman, The Global nonlinear stability of the Minkowski space, Princeton University Press (1993) [INSPIRE].
  51. [51]
    K. Parattu, S. Chakraborty, B.R. Majhi and T. Padmanabhan, A Boundary Term for the Gravitational Action with Null Boundaries, Gen. Rel. Grav. 48 (2016) 94 [arXiv:1501.01053] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  52. [52]
    L. Lehner, R.C. Myers, E. Poisson and R.D. Sorkin, Gravitational action with null boundaries, Phys. Rev. D 94 (2016) 084046 [arXiv:1609.00207] [INSPIRE].
  53. [53]
    W. Wieland, New boundary variables for classical and quantum gravity on a null surface, Class. Quant. Grav. 34 (2017) 215008 [arXiv:1704.07391] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  54. [54]
    C. Duval, G.W. Gibbons and P.A. Horvathy, Conformal Carroll groups and BMS symmetry, Class. Quant. Grav. 31 (2014) 092001 [arXiv:1402.5894] [INSPIRE].
  55. [55]
    P. Kraus, F. Larsen and R. Siebelink, The gravitational action in asymptotically AdS and flat space-times, Nucl. Phys. B 563 (1999) 259 [hep-th/9906127] [INSPIRE].
  56. [56]
    R.B. Mann and D. Marolf, Holographic renormalization of asymptotically flat spacetimes, Class. Quant. Grav. 23 (2006) 2927 [hep-th/0511096] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  57. [57]
    G. Barnich, F. Brandt and M. Henneaux, Local BRST cohomology in the antifield formalism. 1. General theorems, Commun. Math. Phys. 174 (1995) 57 [hep-th/9405109] [INSPIRE].
  58. [58]
    A. Ashtekar and R.O. Hansen, A unified treatment of null and spatial infinity in general relativity. I — Universal structure, asymptotic symmetries and conserved quantities at spatial infinity, J. Math. Phys. 19 (1978) 1542 [INSPIRE].
  59. [59]
    A. Ashtekar, Geometry and Physics of Null Infinity, arXiv:1409.1800 [INSPIRE].
  60. [60]
    R. Geroch, Asymptotic Structure of Space-Time, in Asymptotic Structure of Space-Time, F.P. Esposito and L. Witten eds., Springer (1977), pg. 1.Google Scholar

Copyright information

© The Author(s) 2018

Authors and Affiliations

  • Geoffrey Compère
    • 1
  • Adrien Fiorucci
    • 1
  • Romain Ruzziconi
    • 1
  1. 1.Université Libre de Bruxelles and International Solvay InstitutesBrusselsBelgium

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