On the Black Holes in Alternative Theories of Gravity: The Case of Non-linear Massive Gravity

  • Ivan ArrautEmail author
Part of the Springer Proceedings in Physics book series (SPPHY, volume 170)


It is already known that a positive Cosmological Constant \(\varLambda \) sets the scale \(r_0=\left( \frac{3}{2}r_s r_\varLambda ^2\right) ^{1/3}\), which depending on the mass of the source, can be of astrophysical order of magnitude. This scale was interpreted before as the maximum distance in order to get bound orbits. The same scale corresponds to the static observer position if we want to define the Black Hole temperature in an asymptotically de-Sitter space. \(r_0\) also appears inside the non-linear theory of massive gravity (dRGT) as the Vainshtein radius for the \(\varLambda _3\) version of the theory. I compare the role that this scale plays inside these different scenarios.



The author would like to thank Gia Dvali for a very useful discussion during the Karl Schwarzschild meeting 2013 organized in FIAS, Frankfurt/Germany. This work is supported by MEXT (The Ministry of Education, Culture, Sports, Science and Technology) in Japan and KEK Theory Center.


  1. 1.
    S.L. Ba\(\dot{z}a\acute{n}\)ski, V. Ferrari, Analytic Extension of the Schwarzschild-de Sitter Metric, Il Nuovo Cimento, Vol. 91 B, N. 1, 11 Gennaio (1986)Google Scholar
  2. 2.
    I. Arraut, D. Batic, M. Nowakowski, Comparing two approaches to Hawking radiation of Schwarzschild-de Sitter black holes. Class. Quant. Grav. 26, 125006 (2009)MathSciNetCrossRefADSGoogle Scholar
  3. 3.
    R. Bousso, S.W. Hawking, Pair creation of black holes during inflation. Phys. Rev. D 54, 6312–6322 (1996)MathSciNetCrossRefADSGoogle Scholar
  4. 4.
    I. Arraut, The Planck scale as a duality of the Cosmological Constant: S-dS and S-AdS thermodynamics from a single expression. arXiv:1205.6905v3 [gr-qc]
  5. 5.
    G.W. Gibbons, S.W. Hawking, Cosmological event horizons, thermodynamics and particle creation. Phy. Rev. D 15, 2738 (1977)MathSciNetCrossRefADSGoogle Scholar
  6. 6.
    Z. Stuchl\(\acute{i}\), P. Slaný Equatorial circular orbits in the Kerr-de Sitter spacetimes. Phys. Rev. D 69, 064001 (2004)Google Scholar
  7. 7.
    A. Balaguera Antolinez, C.G. B\(\ddot{o}\)hmer, M. Nowakowski, Scales set by the cosmological constant. Class. quant. Grav. 23, 485Google Scholar
  8. 8.
    I. Arraut, D. Batic, M. Nowakowski, Velocity and velocity bounds in static spherically symmetric metrics. Cent. Eur. J. Phys. 9, 926 (2011)Google Scholar
  9. 9.
    I. Arraut, On the astrophysical scales set by the Cosmological Constant. arXiv:1305.0475 [gr-qc]
  10. 10.
    Z. Stuchl\(\acute{i}\)k, The motion of test particles in black hole backgrounds with non-zero cosmological constant. Bull. Astron. Inst. Czech. 34, 129 (1983)Google Scholar
  11. 11.
    Z. Stuchl\(\acute{i}\)k, Some properties of the Schwarzschild de-Sitter and Schwarzschild Anti de-Sitter spacetimes. Phys. Rev. D 60, 044006 (1999)Google Scholar
  12. 12.
    E. Babichev, C. Deffayet, An introduction to the Vainshtein mechanism. Class. Quant. Grav. 30, 184001 (2013)MathSciNetCrossRefADSGoogle Scholar
  13. 13.
    G. Chkareuli, D. Pirtskhalava, Vainshtein mechanism in \(\varLambda _3\) theories. Phys. Lett. B 713, 99–103 (2012)CrossRefADSGoogle Scholar
  14. 14.
    P. Ginsparg, M.J. Perry, Nucl. Phys. B 222, 245 (1983)MathSciNetCrossRefADSGoogle Scholar
  15. 15.
    K. Koyama, G. Niz, G. Tasinato, Strong interactions and exact solutions in non-linear massive gravity. Phys. Rev. D 84, 064033 (2011)CrossRefADSGoogle Scholar
  16. 16.
    L. Berezhiani, G. Chkareuli, C. de Rham, G. Gabadadze, A.J. Tolley, On black holes in massive gravity. Phys. Rev. D 85, 044024 (2012)CrossRefADSGoogle Scholar
  17. 17.
    K. Koyama, G. Niz, G. Tasinato, Analytic solutions in non-linear massive gravity. Phys. Rev. Lett. 107, 131101 (2011)CrossRefADSGoogle Scholar
  18. 18.
    F. Sbisa, G. Niz, K. Koyama, G. Tasinato, Characterising vainshtein solutions in massive gravity. Phys. Rev. D 86, 024033 (2012)Google Scholar
  19. 19.
    T.M. Nieuwenhuizen, Exact Schwarzschild-de Sitter black holes in a family of massive gravity models. Phys. Rev. D 84, 024038 (2011)CrossRefADSGoogle Scholar
  20. 20.
    K. Hinterbichler, Theoretical aspects of massive gravity. Rev. Mod. Phys. 84, 671 (2012)CrossRefADSGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Theory Center, Institute of Particle and Nuclear Studies, The High Energy Accelerator Research OrganizationTsukubaJapan

Personalised recommendations