Abstract
We develop the Hamiltonian formalism of bigravity and bimetric theories for the general form of the interaction potential of two metrics. When studying the role of lapse and shift functions in theories with two metrics, we naturally use the Kuchař formalism in which these functions are independent of the choice of the space-time coordinate system. We find conditions on the potential necessary and sufficient for the existence of four first-class constraints. These constraints realize a well-known hypersurface deformation algebra in the framework of the formalism of Dirac brackets constructed on the base of all second-class constraints. Fixing one of the metrics, we obtain a bimetric theory not containing first-class constraints. Conserved quantities corresponding to symmetries of the background metric can then be expressed ultralocally in terms of the metric interaction potential.
Similar content being viewed by others
References
T. Damour and I. Kogan, Phys. Rev. D, 66, 104024 (2002); arXiv:hep-th/0206042v2 (2002).
N. Rosen, Phys. Rev., 57, 147–150, 150–153 (1940); Ann. Phys., 22, 1–11 (1963); Gen. Rel. Grav., 4, 435–447 (1973).
G. D’Amico, C. de Rham, S. Dubovsky, G. Gabadadze, D. Pirtskhalava, and A. J. Tolley, Phys. Rev. D, 84, 124046; arXiv:1108.5231v1 [hep-th] (2011).
C. de Rham, G. Gabadadze, and A. J. Tolley, Phys. Rev. Lett., 106, 231101 (2011); arXiv:1011.1232v2 [hep-th] (2010); Phys. Lett. B, 711, 190–195 (2012); arXiv:1107.3820v1 [hep-th] (2011).
S. F. Hassan and R. A. Rosen, Phys. Rev. Lett., 108, 041101 (2012); arXiv:1106.3344v3 [hep-th] (2011); S. F. Hassan, R. A. Rosen, and A. Schmidt-May, JHEP, 1202, 026 (2012); arXiv:1109.3230v2 [hep-th] (2011); S. F. Hassan and R. A. Rosen, JHEP, 1202, 126 (2012); arXiv:1109.3515v2 [hep-th] (2011); 1204, 123 (2012); arXiv:1111.2070v1 [hep-th] (2011).
R. Arnowitt, S. Deser, and Ch. W. Misner, “The dynamics of general relativity,” in: Gravitation: An introduction to Current Research (L. Witten, ed.), Wiley, New York (1962), pp. 227–265; arXiv:gr-qc/0405109v1 (2004).
K. Kuchař, J. Math. Phys., 17, 777–791, 792–800, 801–820 (1976); 18, 1589–1597 (1977).
V. O. Solov’ev, Soviet J. Part. Nucl., 19, 482–497 (1988).
P. A. M. Dirac, Lectures on Quantum Mechanics (Belfer Grad. Sch. Sci. Monogr. Ser., Vol. 2), Belfer Graduate School of Science, New York (1964).
D. G. Boulware and S. Deser, Phys. Rev. D, 6, 3368–3382 (1972).
V. A. Rubakov and P. G. Tinyakov, Phys. Usp., 51, 759–792 (2008).
D. Blas, “Aspects of infrared modifications of gravity,” arXiv:0809.3744v1 [hep-th] (2008).
A. Mironov, S. Mironov, A. Morozov, and A. Morozov, “Resolving puzzles of massive gravity with and without violation of Lorentz symmetry,” arXiv:0910.5243v1 [hep-ph] (2009); “Linearized Lorentz-violating gravity and discriminant locus in the moduli space of mass terms,” arXiv:0910.5245v1 [hep-th] (2009).
K. Hinterbichler, Rev. Modern Phys., 84, 671–710 (2012); arXiv:1105.3735v2 [hep-th] (2011).
C. J. Isham, A. Salam, and J. Strathdee, Phys. Lett. B, 31, 300–302 (1970); Phys. Rev. D, 3, 867–873 (1971).
J. Kluson, “Hamiltonian formalism of particular bimetric gravity model,” arXiv:1211.6267v1 [hep-th] (2012).
J. Kluson, “Hamiltonian Formalism of general bimetric gravity,” arXiv:1303.1652v2 [hep-th] (2013).
V. O. Soloviev and M. V. Tchichikina, “Bigravity in Kuchar’s Hamiltonian formalism: 2. The special case,” arXiv:1302.5096v5 [hep-th] (2013).
Author information
Authors and Affiliations
Corresponding author
Additional information
__________
Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 176, No. 3, pp. 393–407, September 2013.
Rights and permissions
About this article
Cite this article
Soloviev, V.O., Chichikina, M.V. Bigravity in the Kuchař Hamiltonian formalism: The general case. Theor Math Phys 176, 1163–1175 (2013). https://doi.org/10.1007/s11232-013-0097-y
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11232-013-0097-y