Skip to main content
SpringerLink
Log in

Algebra of implicitly defined constraints for gravity as the general form of embedding theory

  • Published:
Gravitation and Cosmology Aims and scope Submit manuscript

Abstract

We consider the embedding theory, the approach to gravity proposed by Regge and Teitelboim, in which 4D space-time is treated as a surface in high-dimensional flat ambient space. In its general form, which does not contain artificially imposed constraints, this theory can be viewed as an extension of GR. In the present paper we study the canonical description of the embedding theory in this general form. In this case, one of the natural constraints cannot be written explicitly, in contrast to the case where additional Einsteinian constraints are imposed. Nevertheless, it is possible to calculate all Poisson brackets with this constraint. We prove that the algebra of four emerging constraints is closed, i.e., all of them are first-class constraints. The explicit form of this algebra is also obtained.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

References

  1. T. Regge and C. Teitelboim, in Proceedings of the First Marcel Grossmann Meeting, Trieste, Italy, 1975, Ed. by R. Ruffini (North Holland, Amsterdam, 1977); arXiv: 1612.05256.

  2. A. Friedman, J. Math. Mech. 10, 625 (1961).

    MathSciNet  Google Scholar 

  3. S. Deser, F. A. E. Pirani, and D. C. Robinson, Phys. Rev. D 14, 3301 (1976).

    Article  ADS  Google Scholar 

  4. M. Pavsic, Phys. Lett. A 107, 66 (1985).

    Article  ADS  MathSciNet  Google Scholar 

  5. V. Tapia, Class. Quantum Grav. 6, L49 (1989).

    Article  ADS  Google Scholar 

  6. M. D. Maia, Class. Quantum Grav. 6, 173 (1989).

    Article  ADS  Google Scholar 

  7. I. A. Bandos, Mod. Phys. Lett. A12, 799 (1997); hepth/9608093.

    Article  ADS  Google Scholar 

  8. F. B. Estabrook, R. S. Robinson, and H. RWahlquist, Class. Quantum Grav. 16, 911 (1999).

    Article  ADS  Google Scholar 

  9. D. Karasik and A. Davidson, Phys. Rev. D 67, 064012 (2003); gr-qc/0207061.

    Article  ADS  MathSciNet  Google Scholar 

  10. S. A. Paston and V. A. Franke, Theor. Math. Phys. 153, 1582 (2007); arXiv: 0711.0576.

    Article  Google Scholar 

  11. L. D. Faddeev, Theor. Math. Phys. 166, 279 (2011); arXiv: 0906.4639, arXiv: 0911.0282, arXiv: 1003.2311.

    Article  Google Scholar 

  12. S. A. Paston, Theor. Math. Phys. 169, 1600 (2011); arXiv: 1111.1104.

    Article  MathSciNet  Google Scholar 

  13. S. Willison, “A re-examination of the isometric embedding approach to general relativity,” arXiv: 1311.6203.

  14. R. Maartens and K. Koyama, Living Reviews in Relativity 13, 5 (2010).

    Article  ADS  Google Scholar 

  15. S. Deser and O. Levin, Phys. Rev. D 59, 064004 (1999); hep-th/9809159.

    Article  ADS  MathSciNet  Google Scholar 

  16. R. Banerjee and B. R. Majhi, Phys. Lett. B 690, 83 (2010); arXiv: 1002.0985.

    Article  ADS  MathSciNet  Google Scholar 

  17. S. A. Paston, Class. Quantum Grav. 32, 145009 (2015); arXiv: 1411.4329.

    Article  ADS  MathSciNet  Google Scholar 

  18. C. D. Collinson, J. Math. Phys. 9, 403 (1968).

    Article  ADS  Google Scholar 

  19. A. Davidson and U. Paz, Found. Phys. 30, 785 (2000).

    Article  MathSciNet  Google Scholar 

  20. S. A. Paston and A. A. Sheykin, Class. Quantum Grav. 29, 095022 (2012); arXiv: 1202.1204.

    Article  ADS  Google Scholar 

  21. S. A. Paston and A. A. Sheykin, SIGMA 10, 003 (2014); arXiv: 1304:6550.

    Google Scholar 

  22. S. Carlip, Rept. Prog. Phys. 64, 885 (2001); grqc/0108040.

    Article  ADS  Google Scholar 

  23. S. A. Paston and E. N. Semenova, Grav. Cosmol. 21, 181 (2015); arXiv: 1509.01529.

    Article  ADS  Google Scholar 

  24. A. Davidson, Class. Quantum Grav. 16, 653 (1999); gr-qc/9710005.

    Article  ADS  Google Scholar 

  25. A. Davidson, D. Karasik, and Y. Lederer, “Cold dark matter from dark energy,” gr-qc/0111107

  26. S. A. Paston and A. A. Sheykin, Int. J. Mod. Phys. D 21 (5), 1250043 (2012); arXiv: 1106.5212.

    Article  ADS  Google Scholar 

  27. R. Arnowitt, S. Deser, and C. Misner, in Gravitation: an Introduction to Current Research Ed. by L. Witten (Wiley, New York, 1962); gr-qc/0405109.

  28. M. Ferraris and M. Francaviglia, J. Math. Phys. 26, 1243 (1985).

    Article  ADS  MathSciNet  Google Scholar 

  29. R. Capovilla, J. Guven, and E. Rojas, Class. Quantum Grav. 21, 5563 (2004); hep-th/0404178.

    Article  ADS  Google Scholar 

  30. R. Capovilla, A. Escalante, J. Guven, and E. Rojas, Int. J. Theor. Phys. 48, 2486 (2009); gr-qc/0603126.

    Article  Google Scholar 

  31. R. Cordero, A. Molgado, and E. Rojas, Phys. Rev. D 79, 024024 (2009); arXiv: 0901.1938.

    Article  ADS  MathSciNet  Google Scholar 

  32. R. Cordero, M. Cruz, A. Molgado, and E. Rojas, Gen. Rel. Grav. 46, 1761 (2014); arXiv: 1309.3031.

    Article  ADS  Google Scholar 

  33. V. A. Franke and V. Tapia, Nuovo Cimento B 107, 611 (1992).

    Article  ADS  Google Scholar 

  34. S. A. Paston and A. N. Semenova, Int. J. Theor. Phys. 49, 2648 (2010); arXiv: 1003.0172.

    Article  Google Scholar 

  35. K. Cahill, “Path integrals for actions that are not quadratic in their time derivatives,” arXiv: 1501.00473

Download references

Author information

Authors and Affiliations

  1. Saint-Petersburg State University, Peterhof, Ulyanovskaya ul. 1, St. Petersburg, Russia

    S. A. Paston, E. N. Semenova, V. A. Franke & A. A. Sheykin

Authors
  1. S. A. Paston
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. E. N. Semenova
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. V. A. Franke
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. A. A. Sheykin
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Paston, S.A., Semenova, E.N., Franke, V.A. et al. Algebra of implicitly defined constraints for gravity as the general form of embedding theory. Gravit. Cosmol. 23, 1–7 (2017). https://doi.org/10.1134/S0202289317010145

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0202289317010145

Search

Navigation