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

TL;DR

This paper explores a Hamiltonian framework linking spacetime geometry modifications, scalar and gauge fields, to theories like MOND and TeVeS, potentially explaining galactic dynamics without dark matter.

Contribution

It introduces a Hamiltonian-based conformal mapping approach connecting relativistic geodesics with modified metrics and gauge fields, integrating Kaluza-Klein and TeVeS structures.

Findings

Conformal Hamiltonian maps can model MOND-like modifications.

Gauge transformations preserve Bekenstein-Sanders condition.

Yang-Mills nonlinearities may prevent caustic singularities.

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. 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 ${\cal U}_\mu$ which preserve the Bekenstein-Sanders condition ${\cal U}_\mu {\cal U}^\mu = -1$. 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 nonlinear terms which may avoid the caustic singularity found by Contaldi, et al.