Models of a modified-inertia formulation of MOND
Mordehai Milgrom
Abstract
Models of ``modified-inertia'' formulation of MOND are described and applied to nonrelativistic many-body systems. Whereas the interbody forces are Newtonian, the expression for their inertia is modified from the Newtonian $m\mathbf{a}$ 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 ``modified-gravity'' 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 $\ensuremath{\mu}(\ensuremath{\theta}⟨{a}_{ex}⟩/{a}_{0})$, where $\ensuremath{\mu}(x)$ is the MOND ``interpolating function'' that describes rotation curves, compared with $\ensuremath{\mu}({a}_{ex}/{a}_{0})$ for presently known modified-gravity formulations. The two main differences are that, while ${a}_{ex}$ is the momentary value of the external acceleration, $⟨{a}_{ex}⟩$ is a certain time average of it and that $\ensuremath{\theta}>1$ is an extra factor that depends on the frequency ratio of the external- and internal-field variations. Only ratios of frequencies enter, and ${a}_{0}$ 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 disk galaxy), the first difference disappears, since $⟨{a}_{ex}⟩={a}_{ex}$. But the $\ensuremath{\theta}$ factor can appreciably enhance the EFE in quenching MOND effects, over what is deduced in modified gravity. This $\ensuremath{\theta}$ 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.