Derivation of MOND from Hossenfelder-Verlinde gravity

Verlinde proposed emergent gravity, which naturally explains the Tully-Fisher relation, an empirical relation in galaxy rotation curves. Inspired by this theory, Hossenfelder constructed a covariant formulation of Verlinde's emergent gravity. In this work, we show that the equation of motion gains an extra acceleration in addition to the usual geodesic equation, according to Hossenfelder's theory. Moreover, we show that the extra acceleration is precisely the square root of the Newtonian gravitational acceleration if the mass of the imposter field is negligible, thus completing the proof that Hossenfelder's theory reduces to modified Newtonian dynamics (MOND) and determining which version of MOND it reduces to. We also obtain the value of L in Hossenfelder-Verlinde gravity theory, which is a constant, contrary to what Hossenfelder claimed. Finally, we suggest how the Newtonian limit that suitably describes our observations in Solar System is recovered in Hossenfelder's theory, by considering the mass of the imposter field.