Why the Deep MOND Regime is Not Simply Newtonian Acceleration Plus a Constant
The deep Modified Newtonian Dynamics (MOND) regime fundamentally differs from simply adding a constant acceleration to Newtonian dynamics because MOND aims to address the discrepancies in galactic rotation curves and gravitational dynamics without invoking dark matter. In the deep MOND regime, the characteristic behavior arises at extremely low accelerations, below a critical threshold $$a_0 \approx 1.2 \times 10^{-10} \, \text{m/s}^2.$$ Unlike the Newtonian model, which predicts gravitational acceleration $$a = \frac{GM}{r^2},$$ the MOND framework introduces a modification such that at accelerations $$a \ll a_0,$$ the effective acceleration $a$ scales as $$a = \sqrt{a_0 a_N},$$ where $a_N$ is the Newtonian acceleration and $a_0$ is the critical acceleration parameter introduced by MOND (Milgrom, 1983). This implies that the relationship between acceleration and distance is not linear and does not simply involve adding a constant term to the Newtonian force.
If the deep MOND regime were simply Newtonian acceleration plus a constant, one would expect the force law to follow $$a_\mathrm{eff} = a_N + c,$$ where $c$ is a constant acceleration. This would not lead to the observed asymptotic flatness of rotation curves in galaxies, where the orbital velocity $v$ becomes constant at large radii. In contrast, MOND predicts that at large distances, where $$a \ll a_0,$$ the acceleration $$a \approx \sqrt{GMa_0}/r,$$ which yields a constant rotational velocity $$v^4 = GMa_0.$$ This behavior matches observations, unlike the unrealistic scenario that would arise from merely adding a constant acceleration: the resulting dynamics would suggest a linear increase in force at large distances, inconsistent with the gravitational behavior in galactic outskirts (Sanders & McGaugh, 2002).
Therefore, the deep MOND regime embodies a fundamentally different modification of gravity where the acceleration transitions smoothly from the Newtonian $a_N$ to the MONDian $\sqrt{a_0 a_N}$. This transition respects the symmetries and invariances that a simple additive constant would violate. Additionally, MOND's behavior is derived from an action principle and is consistent with scale invariance at low accelerations, a property absent in a Newtonian-plus-constant framework (Famaey & McGaugh, 2012). Thus, the deep MOND regime’s acceleration cannot be reduced to Newtonian acceleration plus a constant but is a unique, non-linear theory that correctly captures galactic-scale gravitational phenomena.
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