 #### A note on modified inertia theories

Milgrom made the following statement in a discussion about the external field effect and its vector orientation, which in general is hard to know. It has wider relevance, so is recorded here.

In modified inertia theories there is not even an acceleration field defined. In fact different masses can have different accelerations at the same position depending on their orbit*. This is similar to what happens in special relativity, where, as an example, the acceleration of electrons at a given position in an electromagnetic field depends on their momentary velocity, because the inertial mass depends on the velocity through the Lorentz factor. In MOND, the limited result from modified inertia is that for circular orbits (in an axisymmetric potential) we have μ(V2/R)·V2/R=gN, where μ is a function (universal for a given theory) that is derived from the restriction of the kinetic action to circular orbits. `Luckily', this applies nicely to rotation curves** of isolated discs. When an external field is present you cannot apply this relation anymore to the overall motion. And, it certainly does not follow from modified inertia that for the external acceleration itself we have μ(ge)·ge=gNe, as this acceleration, due, e.g. to large scale structure, is certainly not associated with a circular motion.

* The trajectory of a particle matters in modified inertia, which is inevitably non-local. Where a particle has been matters as well as where it is instantaneously. Other instantaneous attrubtes besides position might also matter, e.g., the velocity as in special relativity.

** It need not apply, for example, to the vertical motions of particles in a disk, so one does not expect to be able to apply the simple MOND formula to the motions of stars perpindicular to the plane of the Milky Way in the same way as to a rotation curve.

For further discussion of the EFE, see the MOND laws of galactic dynamics.