Arnowitt–Deser–Misner representation and Hamiltonian analysis of covariant renormalizable gravity

  • Masud Chaichian
  • Markku Oksanen
  • Anca TureanuEmail author
Regular Article - Theoretical Physics


We study the recently proposed Covariant Renormalizable Gravity (CRG), which aims to provide a generally covariant ultraviolet completion of general relativity. We obtain a spacetime decomposed form—an Arnowitt–Deser–Misner (ADM) representation—of the CRG action. The action is found to contain time derivatives of the gravitational fields up to fourth order. Some ways to reduce the order of these time derivatives are considered. The resulting action is analyzed using the Hamiltonian formalism, which was originally adapted for constrained theories by Dirac. It is shown that the theory has a consistent set of constraints. It is, however, found that the theory exhibits four propagating physical degrees of freedom. This is one degree of freedom more than in Hořava–Lifshitz (HL) gravity and two more propagating modes than in general relativity. One extra physical degree of freedom has its origin in the higher order nature of the CRG action. The other extra propagating mode is a consequence of a projectability condition similarly as in HL gravity. Some additional gauge symmetry may need to be introduced in order to get rid of the extra gravitational degrees of freedom.


Poisson Bracket Canonical Variable Shift Vector Hamiltonian Constraint Physical Degree 
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  1. 1.
    M. Chaichian, S. Nojiri, S.D. Odintsov, M. Oksanen, A. Tureanu, Modified F(R) Hořava–Lifshitz gravity: a way to accelerating FRW cosmology. Class. Quantum Gravity 27, 185021 (2010). arXiv:1001.4102 [hep-th] MathSciNetADSCrossRefGoogle Scholar
  2. 2.
    S. Carloni, M. Chaichian, S. Nojiri, S.D. Odintsov, M. Oksanen, A. Tureanu, Modified first-order Hořava–Lifshitz gravity: Hamiltonian analysis of the general theory and accelerating FRW cosmology in power-law F(R) model. Phys. Rev. D 82, 065020 (2010). arXiv:1003.3925 [hep-th] ADSCrossRefGoogle Scholar
  3. 3.
    S. Nojiri, S.D. Odintsov, Covariant renormalizable gravity and its FRW cosmology. Phys. Rev. D 81, 043001 (2010). arXiv:0905.4213 [hep-th] ADSCrossRefGoogle Scholar
  4. 4.
    S. Nojiri, S.D. Odintsov, A proposal for covariant renormalizable field theory of gravity. Phys. Lett. B 691, 60 (2010). arXiv:1004.3613 [hep-th] MathSciNetADSCrossRefGoogle Scholar
  5. 5.
    S. Nojiri, S.D. Odintsov, Covariant power-counting renormalizable gravity: Lorentz symmetry breaking and accelerating early-time FRW universe. Phys. Rev. D 83, 023001 (2011). arXiv:1007.4856 [hep-th] ADSCrossRefGoogle Scholar
  6. 6.
    S. Nojiri, S.D. Odintsov, Unified cosmic history in modified gravity: from F(R) theory to Lorentz non-invariant models. Phys. Rep. (doi: 10.1016/j.physrep.2011.04.001). arXiv:1011.0544 [gr-qc]
  7. 7.
    P. Hořava, Quantum gravity at a Lifshitz Point. Phys. Rev. D 79, 084008 (2009). arXiv:0901.3775 [hep-th] MathSciNetADSCrossRefGoogle Scholar
  8. 8.
    T.P. Sotiriou, M. Visser, S. Weinfurtner, Phenomenologically viable Lorentz-Violating quantum gravity. Phys. Rev. Lett. 102, 251601 (2009). arXiv:0904.4464 [hep-th] MathSciNetADSCrossRefGoogle Scholar
  9. 9.
    D. Blas, O. Pujolàs, S. Sibiryakov, A healthy extension of Hořava gravity. Phys. Rev. Lett. 104, 181302 (2010). arXiv:0909.3525 [hep-th] MathSciNetADSCrossRefGoogle Scholar
  10. 10.
    M. Henneaux, A. Kleinschmidt, G.L. Gómez, A dynamical inconsistency of Hořava gravity. Phys. Rev. D 81, 064002 (2010). arXiv:0912.0399 [hep-th] MathSciNetADSCrossRefGoogle Scholar
  11. 11.
    M. Chaichian, M. Oksanen, A. Tureanu, Hamiltonian analysis of non-projectable modified F(R) Hořava–Lifshitz gravity. Phys. Lett. B 693, 404 (2010). arXiv:1006.3235 [hep-th] MathSciNetADSCrossRefGoogle Scholar
  12. 12.
    R.L. Arnowitt, S. Deser, C.W. Misner, The dynamics of general relativity, gr-qc/0405109, originally in Gravitation: An Introduction to Current Research, ed. by L. Witten, John Wiley & Sons Inc., New York, 1962. Republished in Gen. Relativ. Gravit. 40, 1997 (2008)
  13. 13.
    R.M. Wald, General Relativity (University of Chicago Press, Chicago, 1984) zbMATHGoogle Scholar
  14. 14.
    É. Gourgoulhon, 3+1 formalism and bases of numerical relativity. arXiv:gr-qc/0703035
  15. 15.
    M. Ostrogradski, Mem. Ac. St. Petersbourg VI 4, 385 (1850) Google Scholar
  16. 16.
    D.A. Eliezer, R.P. Woodard, The problem of nonlocality in string theory. Nucl. Phys. B 325, 389 (1989) MathSciNetADSCrossRefGoogle Scholar
  17. 17.
    P.A.M. Dirac, Generalized Hamiltonian dynamics. Can. J. Math. 2, 129 (1950) MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    P.A.M. Dirac, Proc. R. Soc. Lond. Ser. A 246, 326 (1958) MathSciNetADSzbMATHCrossRefGoogle Scholar
  19. 19.
    P.A.M. Dirac, Lectures on Quantum Mechanics (Yeshiva University, New York, 1964) Google Scholar
  20. 20.
    J.M. Pons, Ostrogradski’s theorem for higher-order singular Lagrangians. Lett. Math. Phys. 17, 181 (1989) MathSciNetADSzbMATHCrossRefGoogle Scholar
  21. 21.
    J. Llosa, J. Vives, Hamiltonian formalism for nonlocal Lagrangians. J. Math. Phys. 35, 2856 (1994) MathSciNetADSzbMATHCrossRefGoogle Scholar
  22. 22.
    J.Z. Simon, Higher derivative Lagrangians, nonlocality, problems and solutions. Phys. Rev. D 41, 3720 (1990) MathSciNetADSCrossRefGoogle Scholar
  23. 23.
    S. Capozziello, J. Matsumoto, S. Nojiri, S.D. Odintsov, Dark energy from modified gravity with Lagrange multipliers. Phys. Lett. B 693, 198 (2010). arXiv:1004.3691 [hep-th] ADSCrossRefGoogle Scholar
  24. 24.
    J. Klusoň, Hamiltonian analysis of Lagrange multiplier modified gravity. arXiv:1009.6067 [hep-th]
  25. 25.
    R.P. Woodard, Avoiding dark energy with 1/R modifications of gravity. Lect. Notes Phys. 720, 403 (2007). arXiv:astro-ph/0601672 ADSCrossRefGoogle Scholar

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© Springer-Verlag / Società Italiana di Fisica 2011

Authors and Affiliations

  1. 1.Department of PhysicsUniversity of HelsinkiHelsinkiFinland

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