Hamiltonian Map to Conformal Modification of Spacetime Metric: KaluzaKlein and TeVeS
Abstract
It has been shown that the orbits of motion for a wide class of nonrelativistic 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 postNewtonian 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 KaluzaKlein 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}}_{μ}(x) which preserve the BekensteinSanders condition {mathcal{U}}_{μ}{mathcal{U}}^{μ}=1. The underlying quantum structure giving rise to these gauge fields is a Hilbert bundle, and the gauge transformations induce a noncommutative behavior to the fields, i.e. they become of YangMills type. Working in the infinitesimal gauge neighborhood of the initial Abelian theory we show that in the Abelian limit the YangMills field equations provide residual nonlinear terms which may avoid the caustic singularity found by Contaldi et al.
 Publication:

Foundations of Physics
 Pub Date:
 January 2011
 DOI:
 10.1007/s1070101094836
 arXiv:
 arXiv:0907.4242
 Bibcode:
 2011FoPh...41..141H
 Keywords:

 General relativity;
 Dark matter and dark energy;
 Properties of galaxies;
 Relativistic astrophysics;
 Conformal modification of Einstein metric;
 MOND;
 TeVeS;
 KaluzaKlein theory;
 Relativistic noncompact YangMills theory with Lorentz group valued gauge transformations on a Hilbert bundle;
 Mathematical Physics;
 General Relativity and Quantum Cosmology;
 High Energy Physics  Theory;
 Physics  Classical Physics
 EPrint:
 Plain TeX, 8 pages. Proceedings of Conference of International Association for Relativistic Dynamics, Thessaloniki, Greece, June 2008. Revision includes discussion of field norm preserving gauge on Hilbert bundle and nonlinear contributions to field equations in Abelian limit