Dynamics with a Nonstandard Inertia-Acceleration Relation: An Alternative to Dark Matter in Galactic Systems

Abstract

Abstract We investigate particle dynamics that is governed by a nonstandard kinetic action of a special form. We are guided by a phenomenological scheme-the modified dynamics (MOND)-that imputes the mass discrepancy, observed in galactic systems, not to the presence of dark matter, but to a departure from Newtonian dynamics below a certain scale of accelerations, a 0 . The particle′s equation of motion in a potential φ is derived from an action, S , of the form S ∝ S k [ r ( t ), a 0 ] − ∫ φ dt . The limit a 0 → 0 corresponds to Newtonian dynamics, and there the kinetic action S k must take the standard form. In the opposite limit, a 0 → ∞ we require S k → 0-and more specifically, for circular orbits S k ∝ a −1 0 -in order to attain the phenomenological success of MOND. Galilei-invariant such theories must be strongly nonlocal. This is a blessing, as such theories need not suffer from the illnesses that are endemic to higher-derivative theories. We comment on the possibility that such a modified law of motion is an effective theory resulting from the elimination of degrees of freedom pertaining to the universe at large (the near equality a 0 ≍ cH 0 being a trace of that connection). We derive a general virial relation for bounded trajectories. Exact solutions are obtained for circular orbits, which pertain to rotation curves of disk galaxies. We also explore, in passing, theories that depart from the conventional Newtonian dynamics for very low frequencies.