Unusual square roots in the ghost-free theory of massive gravity

  • Alexey Golovnev
  • Fedor Smirnov
Open Access
Regular Article - Theoretical Physics


A crucial building block of the ghost free massive gravity is the square root function of a matrix. This is a problematic entity from the viewpoint of existence and uniqueness properties. We accurately describe the freedom of choosing a square root of a (non-degenerate) matrix. It has discrete and (in special cases) continuous parts. When continuous freedom is present, the usual perturbation theory in terms of matrices can be critically ill defined for some choices of the square root. We consider the new formulation of massive and bimetric gravity which deals directly with eigenvalues (in disguise of elementary symmetric polynomials) instead of matrices. It allows for a meaningful discussion of perturbation theory in such cases, even though certain non-analytic features arise.


Classical Theories of Gravity Space-Time Symmetries Spacetime Singularities 


Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.


  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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    H. van Dam and M.J.G. Veltman, Massive and massless Yang-Mills and gravitational fields, Nucl. Phys. B 22 (1970) 397.ADSGoogle Scholar
  3. [3]
    V.I. Zakharov, Linearized gravitation theory and the graviton mass, JETP Lett. 12 (1970) 312.ADSGoogle Scholar
  4. [4]
    A.I. Vainshtein, To the problem of nonvanishing gravitation mass, Phys. Lett. 39B (1972) 393 [INSPIRE].ADSCrossRefGoogle Scholar
  5. [5]
    E. Babichev and C. Deffayet, An introduction to the Vainshtein mechanism, Class. Quant. Grav. 30 (2013) 184001 [arXiv:1304.7240] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    D. Boulware and S. Deser, Can gravitation have a finite range?, Phys. Rev. D 6 (1972) 3368.ADSGoogle Scholar
  7. [7]
    C. de Rham and G. Gabadadze, Generalization of the Fierz-Pauli action, Phys. Rev. D 82 (2010) 044020 [arXiv:1007.0443] [INSPIRE].
  8. [8]
    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
  9. [9]
    S.F. Hassan and R.A. Rosen, On non-linear actions for massive gravity, JHEP 07 (2011) 009 [arXiv:1103.6055] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    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].
  11. [11]
    S.F. 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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    S.F. 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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    S.F. Hassan and R.A. Rosen, Bimetric gravity from ghost-free massive gravity, JHEP 02 (2012) 126 [arXiv:1109.3515] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    G. D’Amico et al., Massive cosmologies, Phys. Rev. D 84 (2011) 124046 [arXiv:1108.5231] [INSPIRE].ADSGoogle Scholar
  15. [15]
    M. von Strauss et al., Cosmological solutions in bimetric gravity and their observational tests, JCAP 03 (2012) 042 [arXiv:1111.1655] [INSPIRE].CrossRefGoogle Scholar
  16. [16]
    Y. Akrami et al., Bimetric gravity is cosmologically viable, Phys. Lett. B 748 (2015) 37 [arXiv:1503.07521] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  17. [17]
    E. Babichev et al., Heavy spin-2 dark matter, JCAP 09 (2016) 016 [arXiv:1607.03497] [INSPIRE].ADSCrossRefGoogle Scholar
  18. [18]
    C. de Rham, Massive gravity, Living Rev. Rel. 17 (2014) 7 [arXiv:1401.4173] [INSPIRE].CrossRefzbMATHGoogle Scholar
  19. [19]
    A. Schmidt-May and M. von Strauss, Recent developments in bimetric theory, J. Phys. A 49 (2016) 183001 [arXiv:1512.00021] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  20. [20]
    C. Deffayet, J. Mourad and G. Zahariade, A note on ‘symmetric’ vielbeins in bimetric, massive, perturbative and non perturbative gravities, JHEP 03 (2013) 086 [arXiv:1208.4493] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    P. Gratia, W. Hu and M. Wyman, Self-accelerating massive gravity: how Zweibeins walk through determinant singularities, Class. Quant. Grav. 30 (2013) 184007 [arXiv:1305.2916] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    P. Gratia, W. Hu and M. Wyman, Self-accelerating massive gravity: bimetric determinant singularities, Phys. Rev. D 89 (2014) 027502 [arXiv:1309.5947] [INSPIRE].
  23. [23]
    D. Comelli, M. Crisostomi, K. Koyama, L. Pilo and G. Tasinato, New branches of massive gravity, Phys. Rev. D 91 (2015) 121502 [arXiv:1505.00632] [INSPIRE].ADSGoogle Scholar
  24. [24]
    A. Golovnev, ADM analysis and massive gravity, in the proceedings of the 7th Mathematical Physics Meeting: Summer School and Conference on Modern Mathematical Physics, September 9-19, Belgrade, Serbia (2012), arXiv:1302.0687 [INSPIRE].
  25. [25]
    A. Golovnev and F. Smirnov, Dealing with ghost-free massive gravity without explicit square roots of matrices, Phys. Lett. B 770 (2017) 209 [arXiv:1701.01836] [INSPIRE].ADSCrossRefGoogle Scholar
  26. [26]
    J.P. Serre, Lie algebras and Lie groups, Lecture Notes in mathematics, Springer, Germany (1965).Google Scholar
  27. [27]
    F.R. Gantmacher, The theory of matrices, Chelsea Publishing Company, U.K. (2000).Google Scholar
  28. [28]
    A. Golovnev, On the Hamiltonian analysis of non-linear massive gravity, Phys. Lett. B 707 (2012) 404 [arXiv:1112.2134] [INSPIRE].ADSCrossRefGoogle Scholar
  29. [29]
    M.B. Kocic, The square-root isometry of coupled quadratic spaces, Master Thesis, Stockholm University, Stockholm, Sweden (2014).Google Scholar
  30. [30]
    T. Kugo and N. Ohta, Covariant approach to the no-ghost theorem in massive gravity, PTEP 2014 (2014) 043B04 [arXiv:1401.3873] [INSPIRE].
  31. [31]
    L. Bernard, C. Deffayet, A. Schmidt-May and M. von Strauss, Linear spin-2 fields in most general backgrounds, Phys. Rev. D 93 (2016) 084020 [arXiv:1512.03620] [INSPIRE].
  32. [32]
    L. Bernard, C. Deffayet and M. von Strauss, Consistent massive graviton on arbitrary backgrounds, Phys. Rev. D 91 (2015) 104013 [arXiv:1410.8302] [INSPIRE].ADSMathSciNetGoogle Scholar

Copyright information

© The Author(s) 2017

Authors and Affiliations

  1. 1.Faculty of PhysicsSt. Petersburg State UniversitySaint PetersburgRussia
  2. 2.St. Petersburg National Research University of Information Technologies, Mechanics and OpticsSaint PetersburgRussia

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