Journal of High Energy Physics

, 2012:26 | Cite as

Ghost-free massive gravity with a general reference metric

  • S. F. Hassan
  • Rachel A. Rosen
  • Angnis Schmidt-May


Theories of massive gravity inevitably include an auxiliary reference metric. Generically, they also contain an inconsistency known as the Boulware-Deser ghost. Recently, a family of non-linear massive gravity actions, formulated with a flat reference metric, were proposed and shown to be ghost free at the complete non-linear level. In this paper we consider these non-linear massive gravity actions but now formulated with a general reference metric. We extend the proof of the absence of the Boulware-Deser ghost to this case. The analysis is carried out in the ADM formalism at the complete non-linear level. We show that in these models there always exists a Hamiltonian constraint which, with an associated secondary constraint, eliminates the ghost. This result considerably extends the range of known consistent non-linear massive gravity theories. In addition, these theories can also be used to describe a massive spin-2 field in an arbitrary, fixed gravitational background. We also discuss the positivity of the Hamiltonian.


Classical Theories of Gravity Cosmology of Theories beyond the SM 


  1. [1]
    D.G. Boulware and S. Deser, Can gravitation have a finite range?, Phys. Rev. D 6 (1972) 3368 [INSPIRE].ADSGoogle Scholar
  2. [2]
    D.G. Boulware and S. Deser, Inconsistency of finite range gravitation, Phys. Lett. B 40 (1972) 227 [INSPIRE].ADSGoogle Scholar
  3. [3]
    C. de Rham and G. Gabadadze, Generalization of the Fierz-Pauli action, Phys. Rev. D 82 (2010) 044020 [arXiv:1007.0443] [INSPIRE].ADSGoogle Scholar
  4. [4]
    C. de Rham, G. Gabadadze and A.J. Tolley, Resummation of massive gravity, Phys. Rev. Lett. 106 (2011) 231101 [arXiv:1011.1232] [INSPIRE].ADSCrossRefGoogle Scholar
  5. [5]
    S.F. Hassan and R.A. Rosen, Resolving the ghost problem in non-linear massive gravity, Phys. Rev. Lett. 108 (2012) 041101 [arXiv:1106.3344] [INSPIRE].ADSCrossRefGoogle Scholar
  6. [6]
    S.F. Hassan and R.A. Rosen, On non-linear actions for massive gravity, JHEP 07 (2011) 009 [arXiv:1103.6055] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  7. [7]
    C. de Rham, G. Gabadadze and A.J. Tolley, Comments on (super)luminality, arXiv:1107.0710 [INSPIRE].
  8. [8]
    C. de Rham, G. Gabadadze and A. Tolley, Ghost free massive gravity in the Stückelberg language, arXiv:1107.3820 [INSPIRE].
  9. [9]
    C. de Rham, G. Gabadadze and A.J. Tolley, Helicity decomposition of ghost-free massive gravity, JHEP 11 (2011) 093 [arXiv:1108.4521] [INSPIRE].CrossRefGoogle Scholar
  10. [10]
    G. D’Amico et al., Massive cosmologies, Phys. Rev. D 84 (2011) 124046 [arXiv:1108.5231] [INSPIRE].ADSGoogle Scholar
  11. [11]
    A.E. Gumrukcuoglu, C. Lin and S. Mukohyama, Cosmological perturbations of self-accelerating universe in nonlinear massive gravity, arXiv:1111.4107 [INSPIRE].
  12. [12]
    C.J. Isham, A. Salam and J.A. Strathdee, F-dominance of gravity, Phys. Rev. D 3 (1971) 867 [INSPIRE].MathSciNetADSGoogle Scholar
  13. [13]
    A. Salam and J.A. Strathdee, A class of solutions for the strong gravity equations, Phys. Rev. D 16 (1977) 2668 [INSPIRE].ADSGoogle Scholar
  14. [14]
    M. Fierz, Force-free particles with any spin, Helv. Phys. Acta 12 (1939) 3 [INSPIRE].
  15. [15]
    M. Fierz and W. Pauli, On relativistic wave equations for particles of arbitrary spin in an electromagnetic field, Proc. Roy. Soc. Lond. A 173 (1939) 211 [INSPIRE].MathSciNetADSGoogle Scholar
  16. [16]
    A. Higuchi, Forbidden mass range for spin-2 field theory in de Sitter space-time, Nucl. Phys. B 282 (1987) 397 [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  17. [17]
    S. Deser and A. Waldron, Stability of massive cosmological gravitons, Phys. Lett. B 508 (2001) 347 [hep-th/0103255] [INSPIRE].MathSciNetADSGoogle Scholar
  18. [18]
    M. Porrati, No van Dam-Veltman-Zakharov discontinuity in AdS space, Phys. Lett. B 498 (2001) 92 [hep-th/0011152] [INSPIRE].MathSciNetADSGoogle Scholar
  19. [19]
    I.I. Kogan, S. Mouslopoulos and A. Papazoglou, The m → 0 limit for massive graviton in dS 4 and AdS 4 : how to circumvent the van Dam-Veltman-Zakharov discontinuity, Phys. Lett. B 503 (2001) 173 [hep-th/0011138] [INSPIRE].MathSciNetADSGoogle Scholar
  20. [20]
    L. Grisa and L. Sorbo, Pauli-Fierz gravitons on Friedmann-Robertson-Walker background, Phys. Lett. B 686 (2010) 273 [arXiv:0905.3391] [INSPIRE].MathSciNetADSGoogle Scholar
  21. [21]
    F. Berkhahn, D.D. Dietrich and S. Hofmann, Self-protection of massive cosmological gravitons, JCAP 11 (2010) 018 [arXiv:1008.0644] [INSPIRE].ADSCrossRefGoogle Scholar
  22. [22]
    F. Berkhahn, D.D. Dietrich and S. Hofmann, Consistency of relevant cosmological deformations on all scales, JCAP 09 (2011) 024 [arXiv:1104.2534] [INSPIRE].ADSCrossRefGoogle Scholar
  23. [23]
    R.L. Arnowitt, S. Deser and C.W. Misner, The dynamics of general relativity, gr-qc/0405109 [INSPIRE].
  24. [24]
    S.F. Hassan and R.A. Rosen, Confirmation of the secondary constraint and absence of ghost in massive gravity and bimetric gravity, arXiv:1111.2070 [INSPIRE].
  25. [25]
    E. Babichev, C. Deffayet and R. Ziour, The recovery of general relativity in massive gravity via the Vainshtein mechanism, Phys. Rev. D 82 (2010) 104008 [arXiv:1007.4506] [INSPIRE].ADSGoogle Scholar
  26. [26]
    T.M. Nieuwenhuizen, Exact Schwarzschild-de Sitter black holes in a family of massive gravity models, Phys. Rev. D 84 (2011) 024038 [arXiv:1103.5912] [INSPIRE].ADSGoogle Scholar
  27. [27]
    K. Koyama, G. Niz and G. Tasinato, Analytic solutions in non-linear massive gravity, Phys. Rev. Lett. 107 (2011) 131101 [arXiv:1103.4708] [INSPIRE].ADSCrossRefGoogle Scholar
  28. [28]
    K. Koyama, G. Niz and G. Tasinato, Strong interactions and exact solutions in non-linear massive gravity, Phys. Rev. D 84 (2011) 064033 [arXiv:1104.2143] [INSPIRE].ADSGoogle Scholar
  29. [29]
    N. Arkani-Hamed, H. Georgi and M.D. Schwartz, Effective field theory for massive gravitons and gravity in theory space, Annals Phys. 305 (2003) 96 [hep-th/0210184] [INSPIRE].MathSciNetADSzbMATHCrossRefGoogle Scholar
  30. [30]
    S.F. Hassan and R.A. Rosen, Bimetric gravity from ghost-free massive gravity, arXiv:1109.3515 [INSPIRE].
  31. [31]
    K. Hinterbichler, Theoretical aspects of massive gravity, arXiv:1105.3735 [INSPIRE].
  32. [32]
    S.F. Hassan and A. Schmidt-May, A nonlinear construction of ghost-free massive gravity and bimetric gravity actions, in preparation.Google Scholar
  33. [33]
    P. Creminelli, A. Nicolis, M. Papucci and E. Trincherini, Ghosts in massive gravity, JHEP 09 (2005) 003 [hep-th/0505147] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  34. [34]
    J. Kluson, Note about Hamiltonian structure of non-linear massive gravity, JHEP 01 (2012) 013 [arXiv:1109.3052] [INSPIRE].ADSCrossRefGoogle Scholar
  35. [35]
    T. Regge and C. Teitelboim, Role of surface integrals in the Hamiltonian formulation of general relativity, Annals Phys. 88 (1974) 286 [INSPIRE].MathSciNetADSzbMATHCrossRefGoogle Scholar
  36. [36]
    J.D. Brown and J.W. York, Quasilocal energy and conserved charges derived from the gravitational action, Phys. Rev. D 47 (1993) 1407 [gr-qc/9209012] [INSPIRE].MathSciNetADSGoogle Scholar
  37. [37]
    L. Alberte, A.H. Chamseddine and V. Mukhanov, Massive gravity: exorcising the ghost, JHEP 04 (2011) 004 [arXiv:1011.0183] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  38. [38]
    A.H. Chamseddine and V. Mukhanov, Massive gravity simplified: a quadratic action, JHEP 08 (2011) 091 [arXiv:1106.5868] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  39. [39]
    S. Folkerts, A. Pritzel and N. Wintergerst, On ghosts in theories of self-interacting massive spin-2 particles, arXiv:1107.3157 [INSPIRE].
  40. [40]
    E. Kiritsis, Product CFTs, gravitational cloning, massive gravitons and the space of gravitational duals, JHEP 11 (2006) 049 [hep-th/0608088] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  41. [41]
    O. Aharony, A.B. Clark and A. Karch, The CFT/AdS correspondence, massive gravitons and a connectivity index conjecture, Phys. Rev. D 74 (2006) 086006 [hep-th/0608089] [INSPIRE].MathSciNetADSGoogle Scholar
  42. [42]
    H. van Dam and M.J.G. Veltman, Massive and massless Yang-Mills and gravitational fields, Nucl. Phys. B 22 (1970) 397 [INSPIRE].ADSGoogle Scholar
  43. [43]
    S.F. Hassan, S. Hofmann and M. von Strauss, Brane induced gravity, its ghost and the cosmological constant problem, JCAP 01 (2011) 020 [arXiv:1007.1263] [INSPIRE].ADSCrossRefGoogle Scholar
  44. [44]
    F.C.P. Nunes and G.O. Pires, Extending the Barnes-Rivers operators to D = 3 topological gravity, Phys. Lett. B 301 (1993) 339 [INSPIRE].ADSGoogle Scholar

Copyright information

© SISSA, Trieste, Italy 2012

Authors and Affiliations

  • S. F. Hassan
    • 1
  • Rachel A. Rosen
    • 2
  • Angnis Schmidt-May
    • 1
  1. 1.Department of Physics & The Oskar Klein CentreStockholm University, AlbaNova University CentreStockholmSweden
  2. 2.Physics Department and Institute for StringsCosmology, and Astroparticle Physics, Columbia UniversityNew YorkU.S.A

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