Models of modified-inertia formulation of MOND

Abstract

Models of "modified-inertia" formulation of MOND are described and applied to nonrelativistic many-body systems. They involve time-nonlocal equations of motion. Momentum, angular momentum, and energy are (nonlocally) defined, whose total values are conserved for isolated systems. The models make all the salient MOND predictions. Yet, they differ from existing "modified-gravity" formulations in some second-tier predictions. 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. 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 $\mu (\theta ⟨a_{ex}⟩/a_0)$, where $\mu (x)$ is the MOND `interpolating function' that describes rotation curves, compared with $\mu (a_{ex}/a_0)$ 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 $\theta >1$ 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 a system on a circular orbit in a galaxy (such as the vertical dynamics in a disc galaxy), the first difference disappears, since $⟨a_{ex}⟩=a_{ex}$. But the $\theta$ factor can appreciably enhance the EFE in quenching MOND effects, over what is deduced in modified gravity. 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.

14 pages. Version matching the published one. Added a discussion of the initial-value problem, of some uniqueness issues, and of solution by iterations. Added some references

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