Models of modified-inertia formulation of MOND

Abstract

Models of 'modified-inertia' formulation of MOND are described and applied to nonrelativistic many-body systems. Whereas the inter-body forces are Newtonian, the expression for their inertia is modified from the Newtonian ma to comply with the basic tenets of MOND. This results in time-nonlocal equations of motion. Momentum, angular momentum, and energy are (nonlocally) defined for bodies, and the total values are conserved for isolated many-body systems. The models make all the salient MOND predictions. Yet, they differ in important ways from existing 'modifiedgravity' formulations in their second-tier predictions. Indeed, the heuristic value of the model is in limelighting such possible differences. The models describe correctly the motion of a composite body in a low-acceleration field even when the internal accelerations of its constituents are high (e.g., a star in a galaxy). They exhibit a MOND external field effect (EFE) that shows some important differences from what we have come to expect from modified-gravity versions: In one, simple example of the models, what determines the EFE, in the case of a dominant external field, is µ(θ aex /a0), where µ(x) is the MOND 'interpolating function' that describes rotation curves, compared with µ(aex/a0) for presently-known 'modified-gravity' formulations. The two main differences are that while aex is the momentary value of the external acceleration, aex is a certain time average of it, and that θ > 1 is an extra factor that depends on the frequency ratio of the external-and internal-field variations. Only ratios of frequencies enter, and a0 remains the only new dimensioned constant. For example, for a system on a circular orbit in a galaxy (such as the vertical dynamics in a disc galaxy), the first difference disappears, since aex = aex. But the θ factor can appreciably enhance the EFE in quenching MOND effects, over what is deduced in modified gravity. This θ enhancement is important in most applications of the EFE. Some exact solutions are also described, such as for rotation curves, for an harmonic force, and the general, two-body problem, which in the deep-MOND regime reduces to a single-body problem.