Skip to main content
SpringerLink
Log in

Relation between quantum effects in general relativity and embedding theory

  • Published:
Theoretical and Mathematical Physics Aims and scope Submit manuscript

Abstract

We discuss results relevant to the relation between quantum effects in a Riemannian space and on the surface appearing as a result of its isometric embedding in a flat space of a higher dimension. We discuss the correspondence between the Hawking effect fixed by an observer in the Riemannian space with a horizon and the Unruh effect related to an accelerated motion of this observer in the ambient space. We present examples for which this correspondence holds and examples for which there is no correspondence. We describe the general form of the hyperbolic embedding of the metric with a horizon smoothly covering the horizon and prove that there is a correspondence between the Hawking and Unruh effects for this embedding. We also discuss the possibility of relating two-point functions in a Riemannian space and the ambient space in which it is embedded. We obtain restrictions on the geometric parameters of the embedding for which such a relation is known.

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

Similar content being viewed by others

References

  1. N. D. Birrell and P. C. W. Davies, Quantum Fields in Curved Space, Cambridge Univ. Press, Cambridge (1984).

    MATH  Google Scholar 

  2. A. A. Grib, S. G. Mamaev, and V. M. Mostepanenko, Quantum Effects in Strong External Fields [in Russian], Atomizdat, Moscow (1980).

    Google Scholar 

  3. S. W. Hawking, Commun. Math. Phys., 43, 199–220 (1975).

    Article  MathSciNet  ADS  Google Scholar 

  4. W. G. Unruh, Phys. Rev. D, 14, 870–892 (1976).

    Article  ADS  Google Scholar 

  5. A. Friedman, J. Math. Mech., 10, 625–649 (1961).

    MATH  MathSciNet  Google Scholar 

  6. T. Regge and C. Teitelboim, “General relativitya la string: A progress report,” in: Proceedings of the First Marcel Grossmann Meeting (Trieste, Italy, 1975, R. Ruffini, ed.), North Holland, Amsterdam (1977), pp. 77–88.

    Google Scholar 

  7. S. Deser, F. A. E. Pirani, and D. C. Robinson, Phys. Rev. D, 14, 3301–3303 (1976).

    Article  ADS  Google Scholar 

  8. M. Pavšič, Phys. Lett. A, 107, 66–70 (1985).

    Article  MATH  MathSciNet  ADS  Google Scholar 

  9. V. Tapia, Class. Q. Grav., 6, L49–L56 (1989).

    Article  MATH  MathSciNet  ADS  Google Scholar 

  10. M. D. Maia, Class. Q. Grav., 6, 173–183 (1989).

    Article  MATH  MathSciNet  ADS  Google Scholar 

  11. I. A. Bandos, Modern Phys. Lett. A, 12, 799–810 (1997); arXiv:hep-th/9608093v1 (1996).

    Article  MATH  MathSciNet  ADS  Google Scholar 

  12. F. B. Estabrook, R. S. Robinson, and H. R. Wahlquist, Class. Q. Grav., 16, 911–918 (1999).

    Article  MATH  MathSciNet  ADS  Google Scholar 

  13. D. Karasik and A. Davidson, Phys. Rev. D, 67,064012(2003); arXiv:gr-qc/0207061v2 (2002).

    Article  MathSciNet  ADS  Google Scholar 

  14. M. D. Bustamante, F. Debbasch, and M.-E. Brachet, “Classical gravitation as free membrane dynamics,” arXiv:gr-qc/0509090v3 (2005).

    Google Scholar 

  15. S. A. Paston and V. A. Franke, Theor. Math. Phys., 153, 1582–1596 (2007); arXiv:0711.0576v1 [gr-qc] (2007).

    Article  MATH  MathSciNet  Google Scholar 

  16. L. D. Faddeev, Theor. Math. Phys., 166, 279–290 (2011); arXiv:0911.0282v2 [hep-th] (2009)

    Article  MATH  MathSciNet  Google Scholar 

  17. “New action for the Hilbert–Einstein equations,” arXiv:0906.4639v2 [hep-th] (2009)

  18. “3+1 Decomposition in the new action for the Einstein theory of gravitation,” arXiv:1003.2311v1 [gr-qc] (2010).

  19. S. A. Paston and A. A. Sheykin, Internat. J. Mod. Phys. D, 21,1250043(2012); arXiv:1106.5212v3 [gr-qc] (2011).

    Article  MathSciNet  ADS  Google Scholar 

  20. S. Willison, “A re-examination of the isometric embedding approach to general relativity,” arXiv:1311.6203v3 [gr-qc] (2013).

    Google Scholar 

  21. R. Maartens and K. Koyama, Living Reviews in Relativity, 13, lrr-2010-5, http://www.livingreviews.org/lrr-2010-5 (2010).

  22. S. A. Paston, Theor. Math. Phys., 169, 1611–1619 (2011); arXiv:1111.1104v1 [gr-qc] (2011).

    Article  MATH  MathSciNet  Google Scholar 

  23. S. Deser and O. Levin, Class. Q. Grav., 15, L85–L87 (1998); arXiv:hep-th/9806223v2 (1998).

    Article  MATH  MathSciNet  ADS  Google Scholar 

  24. S. Deser and O. Levin, Phys. Rev. D, 59,064004(1999); arXiv:hep-th/9809159v1 (1998).

    Article  MathSciNet  ADS  Google Scholar 

  25. M. Beciu and H. Culetu, Modern Phys. Lett. A, 14, 1–8 (1999).

    Article  MathSciNet  ADS  Google Scholar 

  26. S. A. Paston, Class. Q. Grav., 32,145009(2015); arXiv:1411.4329v2 [gr-qc] (2014).

    Article  MathSciNet  ADS  Google Scholar 

  27. M. Bertola, J. Bros, V. Gorini, U. Moschella, and R. Schaeffer, Nucl. Phys. B, 581, 575–603 (2000); arXiv:hepth/0003098v1 (2000).

    Article  MATH  MathSciNet  ADS  Google Scholar 

  28. Y.-Z. Chu, J. Math. Phys., 55,092501(2014); arXiv:1305.6933v4 [gr-qc] (2013).

    Article  MathSciNet  ADS  Google Scholar 

  29. Y.-Z. Chu, “A line source in Minkowski for the de Sitter spacetime scalar Green’s function: Massive case,” arXiv:1310.2939v3 [gr-qc] (2013).

    Google Scholar 

  30. P. K. Townsend, “Black holes,” arXiv:gr-qc/9707012v1 (1997).

    Google Scholar 

  31. W. Rindler, Am. J. Phys., 34, 1174–1178 (1966).

    Article  ADS  Google Scholar 

  32. J. R. Letaw, Phys. Rev. D, 22, 1345–1351 (1980).

    Article  MathSciNet  ADS  Google Scholar 

  33. E. T. Akhmedov and D. Singleton, JETP Letters, 86, 615–619 (2007); arXiv:0705.2525v3 [hep-th] (2007).

    Article  ADS  Google Scholar 

  34. J. R. Letaw, Phys. Rev. D, 23, 1709–1714 (1981).

    Article  MathSciNet  ADS  Google Scholar 

  35. S. Abdolrahimi, Class. Q. Grav., 31,135009(2014); arXiv:1304.4237v1 [gr-qc] (2013).

    Article  ADS  Google Scholar 

  36. L. C. Barbado and M. Visser, Phys. Rev. D, 86,084011(2012); arXiv:1207.5525v2 [gr-qc] (2012).

    Article  ADS  Google Scholar 

  37. S. Deser and O. Levin, Class. Q. Grav., 14, L163–L168 (1997); arXiv:gr-qc/9706018v1 (1997).

    Article  MathSciNet  ADS  Google Scholar 

  38. C. Fronsdal, Phys. Rev., 116, 778–781 (1959).

    Article  MathSciNet  ADS  Google Scholar 

  39. Y.-W. Kim, Y.-J. Park, and K.-S. Soh, Phys. Rev. D, 62,104020(2000); arXiv:gr-qc/0001045v2 (2000).

    Article  MathSciNet  ADS  Google Scholar 

  40. N. L. Santos, O. J. C. Dias, and J. P. S. Lemos, Phys. Rev. D, 70,124033(2004); arXiv:hep-th/0412076v1 (2004).

    Article  MathSciNet  ADS  Google Scholar 

  41. R.-G. Cai and Y. S. Myung, Phys. Rev. D, 83,107502(2011); arXiv:1012.5709v1 [hep-th] (2010).

    Article  ADS  Google Scholar 

  42. S.-T. Hong, W. T. Kim, J. J. Oh, and Y.-J. Park, Phys. Rev. D, 63,127502(2001); arXiv:hep-th/0103036v1 (2001).

    Article  MathSciNet  ADS  Google Scholar 

  43. S.-T. Hong and S.-W. Kim, Modern Phys. Lett. A, 21, 789–794 (2006); arXiv:gr-qc/0303059v2 (2003).

    Article  MATH  ADS  Google Scholar 

  44. Y.-W. Kim, J. Choi, and Y.-J. Park, Phys. Rev. D, 89,044004(2014); arXiv:1311.0592v2 [gr-qc] (2013).

    Article  ADS  Google Scholar 

  45. H.-Z. Chen, Y. Tian, Y.-H. Gao, and X.-C. Song, JHEP, 0410,011(2004); arXiv:gr-qc/0409107v3 (2004).

    Article  MathSciNet  ADS  Google Scholar 

  46. E. J. Brynjolfsson and L. Thorlacius, JHEP, 0809,066(2008); arXiv:0805.1876v3 [hep-th] (2008).

    Google Scholar 

  47. R. Banerjee, B. R. Majhi, and D. Roy, “Corrections to Unruh effect in tunneling formalism and mapping with Hawking effect,” arXiv:0901.0466v1 [hep-th] (2009).

    Google Scholar 

  48. R. Banerjee and B. R. Majhi, Phys. Lett. B, 690, 83–86 (2010); arXiv:1002.0985v2 [gr-qc] (2010).

    Article  MathSciNet  ADS  Google Scholar 

  49. S. A. Paston and A. A. Sheykin, SIGMA, 10,003(2014); arXiv:1304:6550v3 [gr-qc] (2013).

    MathSciNet  Google Scholar 

  50. S. A. Paston, JHEP, 1406,122(2014); arXiv:1402.3975v2 [gr-qc] (2014).

    Article  MathSciNet  ADS  Google Scholar 

  51. A. Davidson and U. Paz, Found. Phys., 30, 785–794 (2000).

    Article  MathSciNet  Google Scholar 

  52. S. A. Paston and A. A. Sheykin, Class. Q. Grav., 29,095022(2012); arXiv:1202.1204v1 [gr-qc] (2012).

    Article  MathSciNet  ADS  Google Scholar 

  53. A. A. Sheykin, D. A. Grad, and S. A. Paston, PoS (QFTHEP2013), 091; arXiv:1401.7820v1 [gr-qc] (2014).

    Google Scholar 

  54. S. A. Paston, Gravitation as a Theory of Embedding, LAP Lambert Academic Publ., Saarbrücken (2012).

    Google Scholar 

  55. M. Bertola and D. Gouthier, Bol. Soc. Brasil. Mat., n.s., 32, 45–62 (2001).

    Article  MATH  MathSciNet  Google Scholar 

  56. S. A. Paston and A. A. Sheykin, Theor. Math. Phys., 175, 806–815 (2013); arXiv:1306.4826v1 [gr-qc] (2013).

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. St. Petersburg State University, St. Petersburg, Russia

    S. A. Paston

Authors
  1. S. A. Paston
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. A. Paston.

Additional information

This research was supported by St. Petersburg State University (Grant No. 11.38.660.2013).

Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 185, No. 1, pp. 162–178, October, 2015.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Paston, S.A. Relation between quantum effects in general relativity and embedding theory. Theor Math Phys 185, 1502–1515 (2015). https://doi.org/10.1007/s11232-015-0359-y

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11232-015-0359-y

Keywords

Search

Navigation