Foundations of Physics

, Volume 41, Issue 1, pp 141–157 | Cite as

Hamiltonian Map to Conformal Modification of Spacetime Metric: Kaluza-Klein and TeVeS

  • Lawrence Horwitz
  • Avi GershonEmail author
  • Marcelo Schiffer


It has been shown that the orbits of motion for a wide class of non-relativistic Hamiltonian systems can be described as geodesic flows on a manifold and an associated dual by means of a conformal map. This method can be applied to a four dimensional manifold of orbits in spacetime associated with a relativistic system. We show that a relativistic Hamiltonian which generates Einstein geodesics, with the addition of a world scalar field, can be put into correspondence in this way with another Hamiltonian with conformally modified metric. Such a construction could account for part of the requirements of Bekenstein for achieving the MOND theory of Milgrom in the post-Newtonian limit. The constraints on the MOND theory imposed by the galactic rotation curves, through this correspondence, would then imply constraints on the structure of the world scalar field. We then use the fact that a Hamiltonian with vector gauge fields results, through such a conformal map, in a Kaluza-Klein type theory, and indicate how the TeVeS structure of Bekenstein and Saunders can be put into this framework. We exhibit a class of infinitesimal gauge transformations on the gauge fields \({\mathcal{U}}_{\mu}(x)\) which preserve the Bekenstein-Sanders condition \({\mathcal{U}}_{\mu}{\mathcal{U}}^{\mu}=-1\). The underlying quantum structure giving rise to these gauge fields is a Hilbert bundle, and the gauge transformations induce a non-commutative behavior to the fields, i.e. they become of Yang-Mills type. Working in the infinitesimal gauge neighborhood of the initial Abelian theory we show that in the Abelian limit the Yang-Mills field equations provide residual nonlinear terms which may avoid the caustic singularity found by Contaldi et al.


General relativity Dark matter and dark energy Properties of galaxies Relativistic astrophysics Conformal modification of Einstein metric MOND TeVeS Kaluza-Klein theory Relativistic noncompact Yang-Mills theory with Lorentz group valued gauge transformations on a Hilbert bundle 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Misner, C.W., Thorne, K.S., Wheeler, J.A.: Gravitation. Freeman, San Francisco (1973) Google Scholar
  2. 2.
    Stueckelberg, E.C.G.: Helv. Phys. Acta 14, 372, 588 (1941) MathSciNetGoogle Scholar
  3. 3.
    Stueckelberg, E.C.G.: Helv. Phys. Acta 15, 23 (1942) zbMATHMathSciNetGoogle Scholar
  4. 4.
    Horwitz, L.P., Piron, C.: Helv. Phys. Acta 46, 316 (1973) Google Scholar
  5. 5.
    Feynman, R.P.: Phys. Rev. 80, 4401 (1950) CrossRefMathSciNetGoogle Scholar
  6. 6.
    Schwinger, J.S.: Phys. Rev. 82, 664 (1951) zbMATHCrossRefMathSciNetADSGoogle Scholar
  7. 7.
    Oron, O., Horwitz, L.P.: Found. Phys. 31, 951 (2001) CrossRefMathSciNetGoogle Scholar
  8. 8.
    Saad, D., Horwitz, L.P., Arshansky, R.I.: Found. Phys. 19, 1125 (1989) CrossRefMathSciNetADSGoogle Scholar
  9. 9.
    Shnerb, N., Horwitz, L.P.: Phys. Rev. A 48, 4068 (1993) CrossRefMathSciNetADSGoogle Scholar
  10. 10.
    Horwitz, L.P., Levitan, J., Lewkowicz, M., Shiffer, M., Zion, Y. Ben: Phys. Rev. Lett. 98, 234301 (2007) CrossRefADSGoogle Scholar
  11. 11.
    Bekenstein, J.D., Milgrom, M.: Astrophys. J. 286, 7 (1984) CrossRefADSGoogle Scholar
  12. 12.
    Bekenstein, J.D.: In: Coley A., Dyer C., Tupper T. (eds.) Second Canadian Conference on General Relativity and Relativistic Astrophysics. Singapore, World Scientific (1992) Google Scholar
  13. 13.
    Milgrom, M.: Astrophys. J. 270, 365, 371, 384 (1983) ADSGoogle Scholar
  14. 14.
    Milgrom, M.: Ann. Phys. 229, 384 (1994) CrossRefADSGoogle Scholar
  15. 15.
    Milgrom, M.: Astrophys. J. 287, 571 (1984) CrossRefADSGoogle Scholar
  16. 16.
    Milgrom, M.: Astrophys. J. 302, 617 (1986) CrossRefADSGoogle Scholar
  17. 17.
    Sanders, R.H.: Astrophys. J. 480, 492 (1997) CrossRefADSGoogle Scholar
  18. 18.
    Bekenstein, J.D.: Phys. Rev. D 70, 083509 (2004) CrossRefADSGoogle Scholar
  19. 19.
    Bekenstein, J.D.: Modified gravity vs. dark matter: relativistic theory for MOND. In: 28th Johns Hopkins Workshop on Current Problems in Particle Theory, June 5–8, 2004. arXiv:astro-ph/0412652 (2005)
  20. 20.
    Gershon, A., Horwitz, L.P.: Kaluza-Klein theory as a dynamics in a dual geometry J. Math. Phys. (2010, in print) Google Scholar
  21. 21.
    Yang, C.N., Mills, R.L.: Phys. Rev. 96, 191 (1954) CrossRefMathSciNetADSGoogle Scholar
  22. 22.
    Contaldi, C.R., Wiseman, T., Withers, B.: arXiv:0802.1215
  23. 23.
    Contaldi, C.R., Wiseman, T., Withers, B.: Phys. Rev. D 78, 044034 (2008) CrossRefADSGoogle Scholar
  24. 24.
    Horwitz, L.P., Levitan, J., Yahalom, A., Lewkowicz, M.: Variational calculus for classical dynamics on dual manifolds (2010, in preparation) Google Scholar
  25. 25.
    Calderon, E., Kupferman, R., Shnider, S., Horwitz, L.P., Calderon, E., Thesis, M.Sc.: Geometric formulation of classical dynamics and hamiltonian chaos, Hebrew University, Jerusalem, June 1 (2009, in preparation) Google Scholar
  26. 26.
    Wesson, P.S.: Space-Time-Matter. Singapore, World Scientific (2007) zbMATHGoogle Scholar
  27. 27.
    Wesson, P.S.: Five Dimensional Physics. Singapore, World Scientific (2006) zbMATHGoogle Scholar
  28. 28.
    Liko, T.: Phys. Lett. B 617, 193 (2005) CrossRefADSGoogle Scholar
  29. 29.
    Mavromatos, N.E., Sakellariadou, M.: Phys. Lett. B 652, 97 (2007), for a study of the possibility of deriving the TeVeS theory from string theory CrossRefMathSciNetADSGoogle Scholar
  30. 30.
    Kaluza, T.: Sitz. Oreuss. Akad. Wiss. 33, 966 (1921) Google Scholar
  31. 31.
    Sagi, E.: Phys. Rev. D 80, 044032 (2009), and arXiv:0905.4001 [gr-qc] 11 Jan. 2010 CrossRefADSGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • Lawrence Horwitz
    • 1
    • 2
    • 3
  • Avi Gershon
    • 1
    Email author
  • Marcelo Schiffer
    • 2
  1. 1.School of PhysicsTel Aviv UniversityRamat AvivIsrael
  2. 2.Department of PhysicsAriel University Center of SamariaArielIsrael
  3. 3.Department of PhysicsBar Ilan UniversityRamat GanIsrael

Personalised recommendations