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. 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 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 Sanders can be put into this
framework. We exhibit a class of infinitesimal gauge transformations on the
gauge fields Uμ​ which preserve the Bekenstein-Sanders condition
Uμ​Uμ=−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
nonlinear terms which may avoid the caustic singularity found by Contaldi, et
al.Comment: 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 limi