Determine whether eta equals one permits modified-gravity theories

Determine whether acceptable MOND theories with scaling parameter \(\eta=1\) can modify the gravitational action, rather than reducing to pure modified-inertia theories.

Background

The paper considers mixed modified-gravity–modified-inertia MOND theories parameterized by the deep-MOND scaling dimension $0\leq\eta\leq1$. The endpoint $\eta=0$ corresponds to pure modified gravity, while $\eta=1$ corresponds to pure modified inertia. The author argues that the inverse implication is clear for $\eta=0$, because the deep-MOND kinetic action then cannot depend on the gravitational potential and therefore appears to be Newtonian.

For $\eta=1$, however, the deep-MOND scaling form of the gravitational Lagrangian does not immediately show that the gravitational potential disappears from the kinetic-sector-independent expression. The unresolved issue is therefore whether the endpoint scaling value $\eta=1$ necessarily enforces pure modified inertia or whether acceptable theories at that value can still contain genuinely modified gravity.