Massive gravity: a general analysis

  • D. Comelli
  • F. Nesti
  • L. Pilo


Massive gravity can be described by adding to the Einstein-Hilbert action a function V of metric components. By using the Hamiltonian canonical analysis, we find the most general form of V such that five degrees of freedom propagate non perturbatively. The construction is based on a set of differential equations for V, that remarkably can be solved in terms of two arbitrary functions. Besides recovering the known “Lorentz invariant” massive gravity theory, we find an entirely new class of solutions, with healthy features on the phenomenological side, in particular they are weakly coupled in the solar system and have a high ultraviolet cutoff Λ2 = (mM pl )1/2, where m is the graviton mass scale.


Classical Theories of Gravity Cosmology of Theories beyond the SM 


  1. [1]
    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
  2. [2]
    D. Boulware and S. Deser, Inconsistency of finite range gravitation, Phys. Lett. B 40 (1972) 227 [INSPIRE].ADSGoogle Scholar
  3. [3]
    D. Comelli, M. Crisostomi, F. Nesti and L. Pilo, Degrees of freedom in massive gravity, Phys. Rev. D 86 (2012) 101502 [arXiv:1204.1027] [INSPIRE].ADSGoogle Scholar
  4. [4]
    V.O. Soloviev and M.V. Tchichikina, Bigravity in Kuchars Hamiltonian formalism. 2. The special case, arXiv:1302.5096 [INSPIRE].
  5. [5]
    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
  6. [6]
    S. 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
  7. [7]
    S. Hassan, R.A. Rosen and A. Schmidt-May, Ghost-free massive gravity with a general reference metric, JHEP 02 (2012) 026 [arXiv:1109.3230] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  8. [8]
    S. Hassan and R.A. Rosen, Confirmation of the secondary constraint and absence of ghost in massive gravity and bimetric gravity, JHEP 04 (2012) 123 [arXiv:1111.2070] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  9. [9]
    C. de Rham, G. Gabadadze and A.J. Tolley, Ghost free massive gravity in the Stückelberg language, Phys. Lett. B 711 (2012) 190 [arXiv:1107.3820] [INSPIRE].ADSGoogle Scholar
  10. [10]
    B. Zumino, Effective Lagrangians and broken symmetries, in Lectures on elementary particles and Quantum Field Theory, volume 2, Brandeis Univ., Cambridge U.S.A. (1970), pg. 437 [INSPIRE].
  11. [11]
    H. van Dam and M. Veltman, Massive and massless Yang-Mills and gravitational fields, Nucl. Phys. B 22 (1970) 397 [INSPIRE].ADSGoogle Scholar
  12. [12]
    Y. Iwasaki, Consistency condition for propagators, Phys. Rev. D 2 (1970) 2255 [INSPIRE].ADSGoogle Scholar
  13. [13]
    V.I. Zakharov, Linearized gravitation theory and the graviton mass, JETP Lett. 12 (1971) 312 [Pisma Zh. Eksp. Teor. Fiz. 12 (1970) 447] [INSPIRE].
  14. [14]
    A. Vainshtein, To the problem of nonvanishing gravitation mass, Phys. Lett. B 39 (1972) 393 [INSPIRE].ADSGoogle Scholar
  15. [15]
    E. Babichev, C. Deffayet and R. Ziour, Recovering general relativity from massive gravity, Phys. Rev. Lett. 103 (2009) 201102 [arXiv:0907.4103] [INSPIRE].ADSCrossRefGoogle Scholar
  16. [16]
    N. Kaloper, A. Padilla and N. Tanahashi, Galileon hairs of Dyson spheres, Vainshteins coiffure and Hirsute bubbles, JHEP 10 (2011) 148 [arXiv:1106.4827] [INSPIRE].ADSCrossRefGoogle Scholar
  17. [17]
    G. Chkareuli and D. Pirtskhalava, Vainshtein mechanism in Λ3 -theories, Phys. Lett. B 713 (2012) 99 [arXiv:1105.1783] [INSPIRE].ADSGoogle Scholar
  18. [18]
    C. Burrage, N. Kaloper and A. Padilla, Strong coupling and bounds on the graviton mass in massive gravity, arXiv:1211.6001 [INSPIRE].
  19. [19]
    S. Deser and A. Waldron, Acausality of massive gravity, Phys. Rev. Lett. 110 (2013) 111101 [arXiv:1212.5835] [INSPIRE].ADSCrossRefGoogle Scholar
  20. [20]
    V. Rubakov, Lorentz-violating graviton masses: getting around ghosts, low strong coupling scale and VDVZ discontinuity, hep-th/0407104 [INSPIRE].
  21. [21]
    S. Dubovsky, Phases of massive gravity, JHEP 10 (2004) 076 [hep-th/0409124] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  22. [22]
    V.A. Rubakov and P.G. Tinyakov, Infrared-modified gravities and massive gravitons, Phys. Usp. 51 (2008) 759 [arXiv:0802.4379] [INSPIRE].ADSCrossRefGoogle Scholar
  23. [23]
    D. Blas, D. Comelli, F. Nesti and L. Pilo, Lorentz breaking massive gravity in curved space, Phys. Rev. D 80 (2009) 044025 [arXiv:0905.1699] [INSPIRE].MathSciNetADSGoogle Scholar
  24. [24]
    Z. Berezhiani, D. Comelli, F. Nesti and L. Pilo, Spontaneous Lorentz breaking and massive gravity, Phys. Rev. Lett. 99 (2007) 131101 [hep-th/0703264] [INSPIRE].ADSCrossRefGoogle Scholar
  25. [25]
    G. Gabadadze and L. Grisa, Lorentz-violating massive gauge and gravitational fields, Phys. Lett. B 617 (2005) 124 [hep-th/0412332] [INSPIRE].MathSciNetADSGoogle Scholar
  26. [26]
    L. Grisa, Lorentz-violating massive gravity in curved space, JHEP 11 (2008) 023 [arXiv:0803.1137] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  27. [27]
    D. Comelli, F. Nesti and L. Pilo, Weak massive gravity, arXiv:1302.4447 [INSPIRE].
  28. [28]
    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].MathSciNetADSMATHCrossRefGoogle Scholar
  29. [29]
    C. Isham, A. Salam and J. Strathdee, F-dominance of gravity, Phys. Rev. D 3 (1971) 867 [INSPIRE].MathSciNetADSGoogle Scholar
  30. [30]
    A. Salam and J. Strathdee, A class of solutions for the strong gravity equations, Phys. Rev. D 16 (1977) 2668 [INSPIRE].ADSGoogle Scholar
  31. [31]
    C. Aragone and J. Chela-Flores, Properties of the f-g theory, Nuovo Cim. A 10 (1972) 818 [INSPIRE].ADSCrossRefGoogle Scholar
  32. [32]
    T. Damour and I.I. Kogan, Effective Lagrangians and universality classes of nonlinear bigravity, Phys. Rev. D 66 (2002) 104024 [hep-th/0206042] [INSPIRE].MathSciNetADSGoogle Scholar
  33. [33]
    C.M. Will, The confrontation between general relativity and experiment, Living Rev. Rel. 9 (2006) 3 [gr-qc/0510072] [INSPIRE].Google Scholar
  34. [34]
    L. Pilo, Bigravity as a tool for massive gravity, PoS(EPS-HEP2011)076 [INSPIRE].
  35. [35]
    D. Comelli, M. Crisostomi, F. Nesti and L. Pilo, Spherically symmetric solutions in ghost-free massive gravity, Phys. Rev. D 85 (2012) 024044 [arXiv:1110.4967] [INSPIRE].ADSGoogle Scholar
  36. [36]
    D. Comelli, M. Crisostomi, F. Nesti and L. Pilo, Finite energy for a gravitational potential falling slower than 1/r, Phys. Rev. D 84 (2011) 104026 [arXiv:1105.3010] [INSPIRE].ADSGoogle Scholar
  37. [37]
    M.S. Volkov, Hairy black holes in the ghost-free bigravity theory, Phys. Rev. D 85 (2012) 124043 [arXiv:1202.6682] [INSPIRE].ADSGoogle Scholar
  38. [38]
    Z. Berezhiani, D. Comelli, F. Nesti and L. Pilo, Exact spherically symmetric solutions in massive gravity, JHEP 07 (2008) 130 [arXiv:0803.1687] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  39. [39]
    R.L. Arnowitt, S. Deser and C.W. Misner, The dynamics of general relativity, Gen. Rel. Grav. 40 (2008) 1997 [gr-qc/0405109] [INSPIRE].ADSMATHCrossRefGoogle Scholar
  40. [40]
    M. Henneaux, A. Kleinschmidt and G. Lucena Gomez, Remarks on gauge invariance and first-class constraints, arXiv:1004.3769 [INSPIRE].
  41. [41]
    D. Gilbarg and N.S. Trudinger, Elliptic partial differential equations of second order, Springer-Verlag, Berlin Germany (1983).MATHCrossRefGoogle Scholar
  42. [42]
    A.V. Pogorelov, Monge-Ampère equation, in Encyclopedia of mathematics, M. Hazewinkel ed., Springer, Berlin Germany (2001).Google Scholar
  43. [43]
    D. Fairlie and A. Leznov, General solutions of the Monge-Ampere equation in n-dimensional space, J. Geom. Phys. 16 (1995) 385 [hep-th/9403134] [INSPIRE].MathSciNetADSMATHCrossRefGoogle Scholar
  44. [44]
    Z. Berezhiani, F. Nesti, L. Pilo and N. Rossi, Gravity modification with Yukawa-type potential: dark matter and mirror gravity, JHEP 07 (2009) 083 [arXiv:0902.0144] [INSPIRE].ADSCrossRefGoogle Scholar
  45. [45]
    D. Comelli, F. Nesti and L. Pilo, to appear.Google Scholar

Copyright information

© SISSA, Trieste, Italy 2013

Authors and Affiliations

  1. 1.INFN, Sezione di FerraraFerraraItaly
  2. 2.Gran Sasso Science InstituteL’AquilaItaly
  3. 3.Dipartimento di Scienze Fisiche e ChimicheUniversità di L’AquilaL’AquilaItaly
  4. 4.INFN, Laboratori Nazionali del Gran SassoAssergiItaly

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