Bigravity in Hamiltonian formalism


Cite item

Full Text

Abstract

Theory of bigravity is one of approaches proposed to solve the dark energy problem of the Universe. It deals with two metric tensors, each one is minimally coupled to the corresponding set of matter fields. The bigravity Lagrangian equals to a sum of two General Relativity Lagrangians with the different gravitational coupling constants and different fields of matter accompanied by the ultralocal potential. As a rule, such a theory has 8 gravitational degrees of freedom: the massless graviton, the massive graviton and the ghost. A special choice of the potential, suggested by de Rham, Gabadadze and Toley (dRGT), allows to avoid of the ghost. But the dRGT potential is constructed by means of the matrix square root, and so it is not an explicit function of the metrics components. One way to do with this difficulty is to apply tetrads. Here we consider an alternative approach. The potential as a differentiable function of metrics components is supposed to exist, but we never appeal to the explicit form of this function. Only properties of this function necessary and sufficient to exclude the ghost are studied. The final results are obtained from the constraint analysis and the Poisson brackets calculations. The gravitational variables are the two induced metrics and their conjugated momenta. Also lapse and shift variables for both metrics are involved. After the exclusion of 3 auxiliary variables we stay with 4 first class constraints and 2 second class ones responsible for the ghost exclusion. The requirements for the potential are as follows: 1) the potential should satisfy a system of the first order linear differential equations; 2) the potential should satisfy the homogeneous Monge-Ampere equation in 4 auxiliary variables; 3) the Hessian of the potential in 3 auxiliary variables is non-degenerate.

About the authors

Vladimir O Soloviev

Institute for High Energy Physics, NRC “Kurchatov Institute”

Email: Vladimir.Soloviev@ihep.ru
(Dr. Phys. & Math. Sci.; Vladimir.Soloviev@ihep.ru), Senior Research Associate, Division of Theoretical Physics 1, Ploschad’ Nauki, Protvino, Moskovskaya obl., 142281, Russian Federation

References

  1. Соловьев В. О. Бигравитация в гамильтоновом формализме / Четвертая международная конференция «Математическая физика и ее приложения»: материалы конф.; ред. чл.-корр. РАН И. В. Волович; д.ф.-м.н., проф. В. П. Радченко. Самара: СамГТУ, 2014. С. 334-335.
  2. Rosen N. General Relativity and Flat Space. I // Phys. Rev., 1940. vol. 57, no. 2. pp. 147-150. doi: 10.1103/physrev.57.147.
  3. Rosen N. General Relativity and Flat Space. II // Phys. Rev., 1940. vol. 57, no. 2. pp. 150-153. doi: 10.1103/physrev.57.150.
  4. Rosen N. Flat-space metric in general relativity theory // Ann. of Phys., 1963. vol. 22, no. 1. pp. 1-11. doi: 10.1016/0003-4916(63)90293-8.
  5. Rosen N. A bi-metric theory of gravitation // Gen. Rel. Grav., 1973. vol. 4, no. 6. pp. 435-447. doi: 10.1007/bf01215403.
  6. Isham C. J., Salam A., Strathdee J. Spontaneous breakdown of conformal symmetry // Phys. Lett. B, 1970. vol. 31, no. 5. pp. 300-302. doi: 10.1016/0370-2693(70)90177-2.
  7. Isham C. J., Salam A., Strathdee J. f -Dominance of Gravity // Phys. Rev. D, 1971. vol. 3, no. 4. pp. 867-873. doi: 10.1103/physrevd.3.867.
  8. Zumino B. Effective Lagrangians and broken symmetries / Lectures on Elementary Particles and Quantum Field Theory. vol. 2; eds. S. Deser, M. Grisaru, H. Pedleton. Cambridge, MA: MIT Press, 1970. pp. 437-500.
  9. Damour T., Kogan I. I. Effective Lagrangians and universality classes of nonlinear bigravity // Phys. Rev. D, 2002. vol. 66, no. 10, 104024. 17 pp., arXiv: hep-th/0206042. doi: 10.1103/physrevd.66.104024.
  10. de Rham C., Gabadadze G., Tolley A. J. Resummation of Massive Gravity // Phys. Rev. Lett., 2011. vol. 106, no. 23, 231101. 4 pp., arXiv: 1011.1232 [hep-th]. doi: 10.1103/physrevlett.106.231101.
  11. de Rham C., Gabadadze G., Tolley A. J. Ghost free massive gravity in the Stückelberg language // Phys. Lett. B, 2012. vol. 711, no. 2. pp. 190-195, arXiv: 1107.3820 [hep-th]. doi: 10.1016/j.physletb.2012.03.081.
  12. Boulware D. G., Deser S. Can Gravitation Have a Finite Range? // Phys. Rev. D, 1972. vol. 6, no. 12. pp. 3368-3382. doi: 10.1103/physrevd.6.3368.
  13. Hassan S. F., Rosen R. A. Bimetric gravity from ghost-free massive gravity // J. High Energ. Phys. vol. 2012, no. 2, 126, arXiv: 1109.3515 [hep-th]. doi: 10.1007/jhep02(2012)126.
  14. Hassan S. F., Rosen R. A. Confirmation of the secondary constraint and absence of ghost in massive gravity and bimetric gravity // J. High Energ. Phys., 2012. vol. 2012, no. 4, 123, arXiv: 1111.2070 [hep-th]. doi: 10.1007/jhep04(2012)123.
  15. Hinterbichler K., Rosen R. A. Interacting spin-2 fields // J. High Energ. Phys., 2012. vol. 2012, no. 7, 047, arXiv: 1203.5783 [hep-th]. doi: 10.1007/jhep07(2012)047.
  16. Alexandrov S., Krasnov K., Speziale S. Chiral description of massive gravity // J. High Energ. Phys., 2013. vol. 2013, no. 6, 068, arXiv: 1212.3614 [hep-th]. doi: 10.1007/JHEP06(2013)068.
  17. Alexandrov S. Canonical structure of tetrad bimetric gravity // Gen. Rel. Grav., 2014. vol. 46, no. 1, 1639, arXiv: 1308.6586 [hep-th]. doi: 10.1007/s10714-013-1639-1.
  18. Kluson J. Hamiltonian formalism of bimetric gravity in vierbein formulation // Eur. Phys. J. C. vol. 74, no. 8, 2985, arXiv: 1307.1974 [hep-th]. doi: 10.1140/epjc/s10052-014-2985-1.
  19. Soloviev V. O. Bigravity in tetrad Hamiltonian formalism and matter couplings, 2014. 25 pp., arXiv: 1410.0048 [hep-th].
  20. Соловьев В. О., Чичикина М. В. Бигравитация в гамильтоновом формализме Кухаржа. Общий случай // ТМФ, 2013. Т. 176, № 3. С. 393-407. doi: 10.4213/tmf8450.
  21. Soloviev V. O., Tchichikina M. V. Bigravity in Kuchar's Hamiltonian formalism. 2. The special case // Phys. Rev. D, 2013. vol. 88, no. 8, 084026, arXiv: 1302.5096 [hep-th]. doi: 10.1103/PhysRevD.88.084026.
  22. Comelli D., Crisostomi M., Nesti F., Pilo L. Degrees of freedom in massive gravity // Phys. Rev. D, 2012. vol. 86, no. 10, 101502(R), arXiv: 1204.1027 [hep-th]. doi: 10.1103/physrevd.86.101502.
  23. Comelli D., Nesti F., Pilo L. Weak massive gravity // Phys. Rev. D, 2013. vol. 87, no. 12, arXiv: 1302.4447 [hep-th]. doi: 10.1103/physrevd.87.124021.
  24. Comelli D., Nesti F., Pilo L. Massive gravity: a general analysis // J. High Energ. Phys., 2013. vol. 2013, no. 7, 161, arXiv: 1305.0236 [hep-th]. doi: 10.1007/jhep07(2013)161.
  25. Arnowitt R., Deser S., Misner Ch. W. The Dynamics of General Relativity, Chapter 7 /Gravitation: an introduction to current research; ed. L. Witten: Wiley, 1962. pp. 227-265
  26. Arnowitt R., Deser S., Misner Ch. W. Republication of: The dynamics of general relativity // Gen. Relativ. Gravit. vol. 40, no. 9. pp. 1997-2027, arXiv: gr-qc/0405109. doi: 10.1007/s10714-008-0661-1.
  27. Kuchař K. Geometry of hyperspace. I // J. Math. Phys., 1976. vol. 17, no. 5. pp. 777-791. doi: 10.1063/1.522976.
  28. Kuchař K. Kinematics of tensor fields in hyperspace. II // J. Math. Phys., 1976. vol. 17, no. 5. pp. 792-800. doi: 10.1063/1.522977.
  29. Kuchař K. Dynamics of tensor fields in hyperspace. III // J. Math. Phys., 1976. vol. 17, no. 5. pp. 801-820. doi: 10.1063/1.522978.
  30. Kuchař K. Geometrodynamics with tensor sources. IV // J. Math. Phys., 1977. vol. 18, no. 8. pp. 1589-1597. doi: 10.1063/1.523467.
  31. Fairlie D., Leznov A. General solutions of the Monge-Ampère equation in n-dimensional space // J. Geom. Phys., 1995. vol. 16, no. 4. pp. 385-390, arXiv: hep-th/9403134. doi: 10.1016/0393-0440(94)00035-3.

Supplementary files

Supplementary Files
Action
1. JATS XML
Download

Copyright (c) 2015 Samara State Technical University

Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License.

Согласие на обработку персональных данных

 

Используя сайт https://journals.rcsi.science, я (далее – «Пользователь» или «Субъект персональных данных») даю согласие на обработку персональных данных на этом сайте (текст Согласия) и на обработку персональных данных с помощью сервиса «Яндекс.Метрика» (текст Согласия).