Is MOND necessarily nonlinear?

Abstract

The iconic, deep-MOND-limit (DML) relation between acceleration and mass, $a\sim (M{\mathcal{A}}_0{)}^{1/2}/r$, implies that, in MOND, accelerations cannot be linear in the mass distribution (${\mathcal{A}}_0\equiv G{a}_0$ is the DML constant, and $a_0$ the MOND acceleration). This leads to important idiosyncracies of MOND, such as a breakdown of the strong equivalence principle, and the resulting ``external-field effect''. I show that the DML axioms are, in themselves, consistent with a, possibly unique, nonrelativistic, action-based, linear formulation of the DML. This model suffers from important drawbacks, which may make it unacceptable as a basis for a full-fledged MOND theory. The model is unique among MOND theories propounded to date not only in being linear -- hence not exhibiting an external-field effect, for example -- but in constituting a modification of both Newtonian inertia and Newtonian gravity. This linear and time-local model inspires and begets several, one-parameter families of models. One family employs nonlinear, time-nonlocal kinetic terms, but still linear gravitational-field equations. Other families generalize the DMLs of AQUAL and QUMOND, modifying gravity as well as inertia. All families employ fractional time derivatives and possibly fractional Laplacians. At present, I cannot base some acceptable MOND theory on these models -- for example, I cannot offer a sensible umbrella theory that interpolates between these DML models and Newtonian dynamics. They are, however, quite useful in elucidating various matter-of-principle aspects of MOND; e.g., they help to understand which predictions follow from only the basic tenets of MOND -- so-called primary predictions -- and which are secondary, i.e., theory dependent. The models may also show the way to a wider class of MOND theories. (Abridged.)

21 pages, small changes to match published version

References in corpus (19)

• Modified Newtonian Dynamics (MOND): Observational Phenomenology and Relativistic Extensions
• Ten equivalent definitions of the fractional Laplace operator
• Quasi-linear formulation of MOND
• Dynamics with a non-standard inertia-acceleration relation: an alternative to dark matter
• From galactic bars to the Hubble tension: weighing up the astrophysical evidence for Milgromian gravity
• MOND laws of galactic dynamics
• The MOND limit from space-time scale invariance
• Predictions and Outcomes for the Dynamics of Rotating Galaxies
• MOND vs. dark matter in light of historical parallels
• MOND-like Fractional Laplacian Theory
• Field equations and cosmology for a class of nonlocal metric models of MOND
• Nonlocal Metric Realizations of MOND
• Determining Cosmology for a Nonlocal Realization of MOND
• General virial theorem for modified-gravity MOND
• Extending Friedmann equations using fractional derivatives using a Last Step Modification technique: the case of a matter dominated accelerated expanding Universe
• Tripotential MOND theories
• The road to MOND--a novel perspective
• The virial theorem for action-governed theories
• Central-surface-densities correlation in general MOND theories

Cited by in corpus (1)

• Dark matter heating of Planet 9, and its observational implications