On the equivalence of Jordan and Einstein frames in scale-invariant gravity

  • Massimiliano RinaldiEmail author
Regular Article


In this paper we consider the issue of the classical equivalence of scale-invariant gravity in the Einstein and in the Jordan frames. We first consider the simplest example \( f(R)=R^{2}\) and show explicitly that the equivalence breaks down when dealing with Ricci-flat solutions. We discuss the link with the fact that flat solutions in quadratic gravity have zero energy. We also consider the case of scale-invariant tensor-scalar gravity and general f(R) theories. We argue that all scale-invariant gravity models have Ricci flat solutions in the Jordan frame that cannot be mapped into the Einstein frame. In particular, the Minkowski metric exists only in the Jordan frame. In this sense, the two frames are not equivalent.


  1. 1.
    E.E. Flanagan, Class. Quantum Grav. 21, 3817 (2004)CrossRefGoogle Scholar
  2. 2.
    S. Capozziello, P. Martin-Moruno, C. Rubano, Phys. Lett. B 689, 117 (2010)MathSciNetCrossRefGoogle Scholar
  3. 3.
    V. Faraoni, E. Gunzig, Int. J. Theor. Phys. 38, 217 (1999)CrossRefGoogle Scholar
  4. 4.
    N. Deruelle, M. Sasaki, Springer Proc. Phys. 137, 247 (2011)CrossRefGoogle Scholar
  5. 5.
    M.S. Ruf, C.F. Steinwachs, Phys. Rev. D 97, 044050 (2018)MathSciNetCrossRefGoogle Scholar
  6. 6.
    K. Bamba, G. Cognola, S.D. Odintsov, S. Zerbini, Phys. Rev. D 90, 023525 (2014)CrossRefGoogle Scholar
  7. 7.
    D.G. Boulware, G.T. Horowitz, A. Strominger, Phys. Rev. Lett. 50, 1726 (1983)MathSciNetCrossRefGoogle Scholar
  8. 8.
    M. Rinaldi, L. Vanzo, Phys. Rev. D 94, 024009 (2016)MathSciNetCrossRefGoogle Scholar
  9. 9.
    L. Alvarez-Gaume, A. Kehagias, C. Kounnas, D. Lüst, A. Riotto, Fortsch. Phys. 64, 176 (2016)CrossRefGoogle Scholar
  10. 10.
    C. Kounnas, D. Lüst, N. Toumbas, Fortsch. Phys. 63, 12 (2015)CrossRefGoogle Scholar
  11. 11.
    G. Cognola, M. Rinaldi, L. Vanzo, S. Zerbini, Phys. Rev. D 91, 104004 (2015)MathSciNetCrossRefGoogle Scholar
  12. 12.
    G. Cognola, M. Rinaldi, L. Vanzo, Entropy 17, 5145 (2015)MathSciNetCrossRefGoogle Scholar
  13. 13.
    S. Bhattacharya, A. Lahiri, Phys. Rev. Lett. 99, 201101 (2007)MathSciNetCrossRefGoogle Scholar
  14. 14.
    C.M. Chambers, I.G. Moss, Phys. Rev. Lett. 73, 617 (1994)MathSciNetCrossRefGoogle Scholar
  15. 15.
    G. Marozzi, M. Rinaldi, R. Durrer, Phys. Rev. D 83, 105017 (2011)CrossRefGoogle Scholar
  16. 16.
    A. De Felice, S. Tsujikawa, Living Rev. Relativ. 13, 3 (2010)CrossRefGoogle Scholar
  17. 17.
    M. Calzà, M. Rinaldi, L. Sebastiani, Eur. Phys. J. C 78, 178 (2018)CrossRefGoogle Scholar
  18. 18.
    S. Deser, B. Tekin, Phys. Rev. Lett. 89, 101101 (2002)MathSciNetCrossRefGoogle Scholar
  19. 19.
    S. Deser, B. Tekin, Phys. Rev. D 67, 084009 (2003)MathSciNetCrossRefGoogle Scholar
  20. 20.
    S. Deser, B. Tekin, Phys. Rev. D 75, 084032 (2007)MathSciNetCrossRefGoogle Scholar

Copyright information

© Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of PhysicsUniversity of TrentoTrentoItaly
  2. 2.TIFPA - INFNTrentoItaly

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