Towards an interpretation of MOND as a modification of inertia: Figure 1.

Abstract

We explore the possibility that Milgrom's modified Newtonian dynamics (MOND) is a manifestation of the modification of inertia at small accelerations. Consistent with the Tully–Fisher relation, dynamics in the small acceleration domain may originate from a quartic (cubic) velocity dependence of energy (momentum) whereas gravitational potentials remain linear with respect to mass. The natural framework for this interpretation is Finsler geometry. The simplest static isotropic Finsler metric of a gravitating mass that incorporates the Tully–Fisher relation at small acceleration is associated with a space–time interval that is either a homogeneous quartic root of polynomials of local displacements or a simple root of a rational fraction thereof. We determine the weak field gravitational equation and find that Finsler space–times that produce a Tully–Fisher relation require that the gravitational potential be modified. For an isolated mass, Newton's potential Mr−1 is replaced by Ma0log (r/r0), where a0 is MOND's acceleration scale and r0 is a yet undetermined distance scale. Orbital energy is linear with respect to mass but angular momentum is proportional to M3/4. Asymptotic light deflection resulting from time curvature is similar to that of a singular isothermal sphere implying that space curvature must be the main source of deflection in static Finsler space–times possibly through the presence of the distance scale r0 that appears in the asymptotic form of the gravitational potential. The quartic nature of the Finsler metric hints at the existence of an underlying area metric that describes the effective structure of space–time.