Hairy Black Holes in Theories with Massive Gravitons

  • Mikhail S. VolkovEmail author
Part of the Lecture Notes in Physics book series (LNP, volume 892)


This is a brief survey of the known black hole solutions in the theories of ghost-free bigravity and massive gravity. Various black holes exist in these theories, in particular those supporting a massive graviton hair. However, it seems that solutions which could be astrophysically relevant are the same as in General Relativity, or very close to them. Therefore, the no-hair conjecture essentially applies, and so it would be hard to detect the graviton mass by observing black holes.


Black Hole Black Hole Solution Massive Gravity Cosmological Term Background Black Hole 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



This work was partly supported by the Russian Government Program of Competitive Growth of the Kazan Federal University.


  1. 1.
    R. Ruffini, J.A. Wheeler, Introducing the black hole. Phys. Today 24, 30–41 (1971)ADSCrossRefGoogle Scholar
  2. 2.
    W. Israel, Event horizons in static vacuum space-times. Phys. Rev. 164, 1776–1779 (1967)ADSCrossRefGoogle Scholar
  3. 3.
    B. Carter, Black hole equilibrium states, in Black Holes, ed. by C. DeWitt, B.S. DeWit (Gordon and Breach, New York, 1973)Google Scholar
  4. 4.
    P.O. Mazur, Proof of uniqueness of the Kerr-Newman black hole solution. J. Phys. A15, 3173–3180 (1982)ADSMathSciNetGoogle Scholar
  5. 5.
    J.D. Bekenstein, Transcendence of the law of baryon-number conservation in black hole physics. Phys. Rev. Lett. 28, 452–455 (1972)ADSCrossRefGoogle Scholar
  6. 6.
    J.D. Bekenstein, Nonexistence of baryon number for static black holes. Phys. Rev. D5, 1239–1246 (1972)ADSMathSciNetGoogle Scholar
  7. 7.
    J.D. Bekenstein, Nonexistence of baryon number for black holes. II. Phys. Rev. D5, 2403–2412 (1972)ADSMathSciNetGoogle Scholar
  8. 8.
    J.D. Bekenstein, Novel ‘no scalar hair’ theorem for black holes. Phys. Rev. D51, 6608–6611 (1995)ADSMathSciNetGoogle Scholar
  9. 9.
    A.E. Mayo, J.D. Bekenstein, No hair for spherical black holes: charged and nonminimally coupled scalar field with selfinteraction. Phys. Rev. D54, 5059–5069 (1996)ADSMathSciNetGoogle Scholar
  10. 10.
    J.D. Bekenstein, Black hole hair: 25 - years after (1996) [arXiv:gr-qc/9605059]Google Scholar
  11. 11.
    S. Hod, Stationary scalar clouds around rotating black holes. Phys. Rev. D86, 104026 (2012)ADSGoogle Scholar
  12. 12.
    C.A.R. Herdeiro, E. Radu, Kerr black holes with scalar hair. Phys. Rev. Lett. 112, 221101 (2014)ADSCrossRefGoogle Scholar
  13. 13.
    P.B. Yasskin, Solutions for gravity coupled to massless gauge fields. Phys. Rev. D12, 2212–2217 (1975)ADSMathSciNetGoogle Scholar
  14. 14.
    M.S. Volkov, D.V. Galtsov, Non-Abelian Einstein Yang-Mills black holes. JETP Lett. 50, 346–350 (1989)ADSGoogle Scholar
  15. 15.
    M.S. Volkov, D.V. Galtsov, Black holes in Einstein Yang-Mills theory. Sov. J. Nucl. Phys. 51, 747–753 (1990)MathSciNetGoogle Scholar
  16. 16.
    M.S. Volkov, D.V. Gal’tsov, Gravitating non-Abelian solitons and black holes with Yang-Mills fields. Phys. Rept. 319, 1–83 (1999)Google Scholar
  17. 17.
    B. Kleihaus, J. Kunz, Static axially symmetric Einstein Yang-Mills dilaton solutions. 2. Black hole solutions. Phys. Rev. D57, 6138–6157 (1998)Google Scholar
  18. 18.
    S.S. Gubser, S.S. Pufu, The Gravity dual of a p-wave superconductor. J. High Energy Phys. 0811, 033 (2008)ADSCrossRefMathSciNetGoogle Scholar
  19. 19.
    M. Fierz, W. Pauli, On relativistic wave equations for particles of arbitrary spin in an electromagnetic field. Proc. R. Soc. Lond. A173, 211–232 (1939)ADSCrossRefMathSciNetGoogle Scholar
  20. 20.
    C. de Rham, G. Gabadadze, A.J. Tolley, Resummation of massive gravity. Phys. Rev. Lett. 106, 231101 (2011)ADSCrossRefGoogle Scholar
  21. 21.
    K. Hinterbichler, Theoretical aspects of massive gravity. Rev. Mod. Phys. 84, 671–710 (2012)ADSCrossRefGoogle Scholar
  22. 22.
    C. de Rham, Massive gravity (2014) [arXiv:1401.4173]Google Scholar
  23. 23.
    D.G. Boulware, S. Deser, Can gravitation have a finite range? Phys. Rev. D6, 3368–3382 (1972)ADSGoogle Scholar
  24. 24.
    S.F. Hassan, R.A. Rosen, Resolving the ghost problem in non-linear massive gravity. Phys. Rev. Lett. 108, 041101 (2012)ADSCrossRefGoogle Scholar
  25. 25.
    S.F. Hassan, R.A. Rosen, Confirmation of the secondary constraint and absence of ghost in massive gravity and bimetric gravity. J. High Energy Phys. 1204, 123 (2012)ADSCrossRefMathSciNetGoogle Scholar
  26. 26.
    J. Kluson, Non-linear massive gravity with additional primary constraint and absence of ghosts. Phys. Rev. D86, 044024 (2012)ADSGoogle Scholar
  27. 27.
    D. Comelli, M. Crisostomi, F. Nesti, L. Pilo, Degrees of freedom in massive gravity. Phys. Rev. D86, 101502 (2012)ADSGoogle Scholar
  28. 28.
    D. Comelli, F. Nesti, L. Pilo, Massive gravity: a general analysis (2013) [arXiv:1305.0236]Google Scholar
  29. 29.
    S.F. Hassan, R.A. Rosen, Bimetric gravity from ghost-free massive gravity. J. High Energy Phys. 1202, 126 (2012)ADSCrossRefMathSciNetGoogle Scholar
  30. 30.
    A.G. Riess et al., Observational evidence from supernovae for an accelerating universe and a cosmological constant. Astron. J. 116(3), 1009 (1998)Google Scholar
  31. 31.
    S. Perlmutter et al., Measurements of Omega and Lambda from 42 high-redshift supernovae. Astrophys. J. 517(2), 565 (1999)Google Scholar
  32. 32.
    T. Damour, I.I. Kogan, A. Papazoglou, Nonlinear bigravity and cosmic acceleration. Phys. Rev. D 66, 104025 (2002)ADSCrossRefMathSciNetGoogle Scholar
  33. 33.
    M.S. Volkov, Hairy black holes in the ghost-free bigravity theory. Phys. Rev. D85, 124043 (2012)ADSGoogle Scholar
  34. 34.
    D. Comelli, M. Crisostomi, F. Nesti, L. Pilo, Spherically symmetric solutions in ghost-free massive gravity. Phys. Rev. D85, 024044 (2012)ADSGoogle Scholar
  35. 35.
    R. Brito, V. Cardoso, P. Pani, Black holes with massive graviton hair. Phys. Rev. D88, 064006 (2013)ADSGoogle Scholar
  36. 36.
    C.J. Isham, A. Salam, J.A. Strathdee, F-dominance of gravity. Phys. Rev. D3, 867–873 (1971)MathSciNetGoogle Scholar
  37. 37.
    M.S. Volkov, Self-accelerating cosmologies and hairy black holes in ghost-free bigravity and massive gravity. Class. Quant. Grav. 30, 184009 (2013)ADSCrossRefGoogle Scholar
  38. 38.
    A. Salam, J.A. Strathdee, A class of solutions for the strong gravity equations. Phys. Rev. D16, 2668 (1977)ADSGoogle Scholar
  39. 39.
    C.J. Isham, D. Storey, Exact spherically symmetric classical solutions for the f-g theory of gravity. Phys. Rev. D18, 1047 (1978)ADSMathSciNetGoogle Scholar
  40. 40.
    Z. Berezhiani, D. Comelli, F. Nesti, L. Pilo, Exact spherically symmetric solutions in massive gravity. J. High Energy Phys. 0807, 130 (2008)ADSCrossRefMathSciNetGoogle Scholar
  41. 41.
    E. Babichev, A. Fabbri, A class of charged black hole solutions in massive (bi)gravity. J. High. Energy Phys. 1407, 016 (2014)ADSCrossRefGoogle Scholar
  42. 42.
    Th.M. Nieuwenhuizen, Exact Schwarzschild-de Sitter black holes in a family of massive gravity models. Phys. Rev. D84, 024038 (2011)ADSGoogle Scholar
  43. 43.
    L. Berezhiani, G. Chkareuli, C. de Rham, G. Gabadadze, A.J. Tolley, On black holes in massive gravity. Phys. Rev. D85, 044024 (2012)ADSGoogle Scholar
  44. 44.
    K. Koyama, G. Niz, G. Tasinato, Analytic solutions in non-linear massive gravity. Phys. Rev. Lett. 107, 131101 (2011)ADSCrossRefGoogle Scholar
  45. 45.
    K. Koyama, G. Niz, G. Tasinato, Strong interactions and exact solutions in non-linear massive gravity. Phys. Rev. D84, 064033 (2011)ADSGoogle Scholar
  46. 46.
    Y.-F. Cai, D.A. Easson, C. Gao, E.N. Saridakis, Charged black holes in nonlinear massive gravity. Phys. Rev. D87(6), 064001 (2013)Google Scholar
  47. 47.
    C. Deffayet, T. Jacobson, On horizon structure of bimetric spacetimes. Class. Quant. Grav. 29, 065009 (2012)ADSCrossRefMathSciNetGoogle Scholar
  48. 48.
    A.I. Vainshtein, To the problem of nonvanishing gravitation mass. Phys. Lett. B39, 393–394 (1972)ADSCrossRefGoogle Scholar
  49. 49.
    E. Babichev, C. Deffayet, R. Ziour, Recovering general relativity from massive gravity. Phys. Rev. Lett. 103, 201102 (2009)ADSCrossRefGoogle Scholar
  50. 50.
    E. Babichev, C. Deffayet, R. Ziour, The recovery of general relativity in massive gravity via the Vainshtein mechanism. Phys. Rev. D82, 104008 (2010)ADSGoogle Scholar
  51. 51.
    A. Gruzinov, M. Mirbabayi, Stars and black holes in massive gravity. Phys. Rev. D84, 124019 (2011)ADSGoogle Scholar
  52. 52.
    F. Sbisa, G. Niz, K. Koyama, G. Tasinato, Characterising Vainshtein solutions in massive gravity. Phys. Rev. D86, 024033 (2012)ADSGoogle Scholar
  53. 53.
    E. Babichev, M. Crisostomi, Restoring general relativity in massive bi-gravity theory. Phys. Rev. D88, 084002 (2013)ADSGoogle Scholar
  54. 54.
    E. Babichev, A. Fabbri, Instability of black holes in massive gravity. Class. Quant. Grav. 30, 152001 (2013)ADSCrossRefMathSciNetGoogle Scholar
  55. 55.
    R. Gregory, R. Laflamme, Black strings and p-branes are unstable. Phys. Rev. Lett. 70, 2837–2840 (1993)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  56. 56.
    R. Brito, V. Cardoso, P. Pani, Massive spin-2 fields on black hole spacetimes: instability of the Schwarzschild and Kerr solutions and bounds on the graviton mass. Phys. Rev. D88(2), 023514 (2013)Google Scholar
  57. 57.
    R. Brito, V. Cardoso, P. Pani, Partially massless gravitons do not destroy general relativity black holes. Phys. Rev. D87(12), 124024 (2013)Google Scholar
  58. 58.
    H. Kodama, I. Arraut, Stability of the Schwarzschild-de Sitter black hole in the dRGT massive gravity theory. Progr. Theor. Exp. Phys. 2014, 023E02 (2014)Google Scholar
  59. 59.
    E. Babichev, A. Fabbri, Stability analysis of black holes in massive gravity: a unified treatment. Phys. Rev. D89, 081502 (2014)ADSGoogle Scholar
  60. 60.
    S. Deser, A. Waldron, Non-Einstein source effects in massive gravity. Phys. Rev. D89, 027503 (2014)ADSGoogle Scholar
  61. 61.
    M. Mirbabayi, A. Gruzinov, Black hole discharge in massive electrodynamics and black hole disappearance in massive gravity. Phys. Rev. D88, 064008 (2013)ADSGoogle Scholar
  62. 62.
    A. Nicolis, R. Rattazzi, E. Trincherini, The Galileon as a local modification of gravity. Phys. Rev. D79, 064036 (2009)ADSMathSciNetGoogle Scholar
  63. 63.
    E. Babichev, C. Charmousis, Dressing a black hole with a time-dependent Galileon. J. High. Energy Phys. 1408, 106 (2014)ADSCrossRefGoogle Scholar
  64. 64.
    L. Hui, A. Nicolis, No-hair theorem for the Galileon. Phys. Rev. Lett. 110(24), 241104 (2013)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Laboratoire de Mathématiques et Physique Théorique CNRS-UMR 7350Université de ToursToursFrance
  2. 2.Institut des Hautes Etudes Scientifiques (IHES)Bures-sur-YvetteFrance
  3. 3.Department of General Relativity and GravitationInstitute of Physics, Kazan Federal UniversityKazanRussia

Personalised recommendations