International Journal of Theoretical Physics

, Volume 56, Issue 9, pp 2746–2754

On a Quantum Theory of Relativity

  • Pedro F. González-Díaz
  • Alberto Rozas-Fernández


As expressed in terms of classical coordinates, the inertial spacetime metric that contains quantum corrections deriving from a quantum potential defined from the quantum probability amplitude is obtained to be given as an elliptic integral of the second kind that does not satisfy Lorentz transformations but a generalised invariance quantum group. Based on this result, we introduce a new, alternative procedure to quantise Einstein general relativity where the metric is also given in terms of elliptic integrals and is free from the customary problems of the current quantum models. We apply the procedure to Schwarzschild black holes and briefly analyse the results.


Quantum mechanics Relativity Quantum gravity 


  1. 1.
    Carlip, S., Chiou, D.W., Ni, W.T., Woodard, R.: Int. J. Mod. Phys. D 24(11), 530028 (2015). doi:10.1142/S0218271815300281. arXiv:1507.08194 [gr-qc]CrossRefGoogle Scholar
  2. 2.
    Nojiri, S., Odintsov, S.D.: Int. J. Mod. Phys. A 16, 1015 (2001). doi:10.1142/S0217751X01002968 [hep-th/0009202]ADSCrossRefGoogle Scholar
  3. 3.
    Hawking, S., Penrose, R.: Proc. Roy. Soc. Lond. A 314, 529 (1970). doi:10.1098/rspa.1970.0021 ADSCrossRefGoogle Scholar
  4. 4.
    Hawking, S., Penrose, R.: The Nature of Space and Time. Princeton University Press, Princeton. The Isaac Newton Institute series of lectures (1996)Google Scholar
  5. 5.
    Tsamis, N.C., Woodard, R.P.: Phys. Rev. D 36, 3641 (1987). doi:10.1103/PhysRevD.36.3641 ADSMathSciNetCrossRefGoogle Scholar
  6. 6.
    Friedman, J.L., Jack, I.: Phys. Rev. D 37, 3495 (1988). doi:10.1103/PhysRevD.37.3495 ADSMathSciNetCrossRefGoogle Scholar
  7. 7.
    DeWitt, B.S.: Phys. Rev. 160, 1113–1148 (1967)ADSCrossRefGoogle Scholar
  8. 8.
    Goroff, M.H., Sagnotti, A.: Phys. Lett. 160B, 81 (1985). doi:10.1016/0370-2693(85)91470-4 ADSCrossRefGoogle Scholar
  9. 9.
    Goroff, M.H., Sagnotti, A.: Nucl. Phys. B 266, 709 (1986). doi:10.1016/0550-3213(86)90193-8 ADSCrossRefGoogle Scholar
  10. 10.
    Buchbinder, I.L., Odintsov, S.D., Shapiro, I.L.: Effective Action in Quantum Gravity. IOP, Bristol (1992)Google Scholar
  11. 11.
    Einstein’s miraculous year. In: Stachel, J. (ed.) Five Papers that Changed the Face of Physics. Princeton University Press, Princeton (2005)Google Scholar
  12. 12.
    Einstein, A.: Sit. Preus. Akad. Wiss. Berlin 844 (1915)Google Scholar
  13. 13.
    Isham, C. J.: *Bad honnef Proceedings, Canonical gravity* 1-21. and London Imp. Coll. - ICTP-93-94-01 (93/10,rec.Nov.) 21 p [gr-qc/9310031] (1993)Google Scholar
  14. 14.
    Hawking, S.W.: Phys. Scr., T 15, 151 (1987)ADSCrossRefGoogle Scholar
  15. 15.
    Page, D.N.: Phys. Rev. D 34, 2267 (1986)ADSMathSciNetCrossRefGoogle Scholar
  16. 16.
    Landau, L.D., Lifschitz, E.M.: The classical theory of fields. Pergamon Press, Oxford, UK (1975)Google Scholar
  17. 17.
    Bohm, D.J.: Quantum theory. Dover, Prentice Hall, New York, USA (1951)MATHGoogle Scholar
  18. 18.
    Bohm, D.: Phys. Rev. 85, 166 (1952). Phys. Rev. 85, 180 (1952)ADSCrossRefGoogle Scholar
  19. 19.
    Gonzalez-Diaz, P.F.: Phys. Rev. D 69, 103512 (2004). [astro-ph/0311244]ADSCrossRefGoogle Scholar
  20. 20.
    Abramowitz, M., Stegun, I.A.: Handbook of mathematical functions. Dover, New york, USA (1972)MATHGoogle Scholar
  21. 21.
    Lorentz, H.A.: Proc. Acad. Science Amsterdam 1,427(1899), 4,669 (1904)Google Scholar
  22. 22.
    DeWitt, B.S.: Phys. Rev. 160, 1113 (1967)ADSCrossRefGoogle Scholar
  23. 23.
    Dirac, P.A.M.: Lectures on quantum mechanics. Yeshiva University, New York, USA (1964)MATHGoogle Scholar
  24. 24.
    Schwarzschild, K.: vol. 189. Sit. Deut. Akad. Wissen., Berlin (1916)Google Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  • Pedro F. González-Díaz
    • 1
  • Alberto Rozas-Fernández
    • 2
    • 3
  1. 1.Instituto de Física FundamentalConsejo Superior de Investigaciones CientíficasMadridSpain
  2. 2.Instituto de Astrofísica e Ciências do EspaçoUniversidade de Lisboa, OALLisboaPortugal
  3. 3.Departamento de Física, Faculdade de CiênciasUniversidade de LisboaLisbonPortugal

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