Modified Newtonian Dynamics (MOND)

• MOND is a hypothesis that modifies Newtonian dynamics in low-acceleration regimes to explain flat galactic rotation curves and the baryonic Tully–Fisher relation without invoking dark matter.
• It employs an algebraic interpolating function, such as the standard or simple form, to transition smoothly from Newtonian to MOND behavior based on a characteristic acceleration scale.
• Empirical successes include matching the Radial Acceleration Relation with minimal intrinsic scatter, while challenges persist in galaxy clusters and Solar System tests.

Modified Newtonian Dynamics (MOND) is a hypothesis-driven framework proposing a modification of classical Newtonian gravity and inertia in the regime of low accelerations. Motivated by persistent mass anomalies in galactic rotation curves and scaling relations, MOND introduces a characteristic acceleration scale, $a_0$ (of order $10^{-10}\rm\,m/s^2$), below which gravitational dynamics depart from Newtonian predictions. Rather than requiring an unseen dark matter component, MOND describes galaxy phenomenology as a direct consequence of altered dynamical laws, mapping observed baryonic mass distributions uniquely to gravitational fields. Despite this empirical success, MOND remains a phenomenological, non-relativistic paradigm requiring a fundamental theoretical embedding for full coherence across all astrophysical and cosmological scales (Famaey et al., 28 Jan 2025).

1. The MOND Paradigm: Law and Range of Applicability

MOND replaces Newton’s second law or the Newtonian Poisson equation with an algebraic relation involving an interpolating function $\mu$. In its canonical form (Milgrom 1983), the theory sets

$a\,\mu\left(\frac{a}{a_0}\right) = a_N$

where $a_N=GM/r^2$ is the Newtonian acceleration. The interpolating function $\mu(x)$ enforces the required asymptotics: $$\mu(x)\to1\quad(x\gg1),\quad\mu(x)\to x\quad(x\ll1).$$ Common examples include the “standard” and “simple” forms: $$\mu_\text{std}(x) = \frac{x}{\sqrt{1+x^2}}, \qquad \mu_\text{simp}(x) = \frac{x}{1+x}.$$ In the deep-MOND limit ($a\ll a_0$), the observed acceleration scales as

$a \simeq \sqrt{a_0\,a_N}.$

This algebraic rule immediately yields two cornerstone phenomena: (i) flat galactic rotation curves far from the luminous baryonic center, and (ii) the baryonic Tully–Fisher relation

$V_\infty^4 = G\,M_{\mathrm{bar}}\,a_0,$

observed with near-zero intrinsic scatter over several orders of magnitude in galaxy mass (Famaey et al., 28 Jan 2025, Famaey et al., 2011).

2. Empirical Foundations: Rotation Curves and the Radial Acceleration Relation

MOND predicts rotation curves from baryonic matter alone, fitting both high-surface-brightness and gas-dominated galaxies across a wide mass range using only the fundamental quantity $a_0$ and the mass-to-light ratio as free parameters (Desmond, 27 May 2025). The Radial Acceleration Relation (RAR)—the tight correlation between observed centripetal acceleration and that predicted by baryons,

$$a_{\mathrm{obs}}(r) = \nu\left(\frac{a_N(r)}{a_0}\right)a_N(r)$$

—is naturally recovered. The observed relation is so tight ($\sigma_\mathrm{int} \sim 0.034 \pm 0.002$ dex in SPARC) that it challenges the degree of freedom allowed in ΛCDM unless fine-tuned (Desmond, 27 May 2025, Dutton et al., 2019). Notably, high-resolution cosmological simulations (NIHAO) have shown that the RAR and the emergent acceleration scale can arise in ΛCDM through galaxy formation physics, suggesting that MOND’s empirical foundation might reflect deeper regularities without requiring fundamental modification of gravity (Dutton et al., 2019).

3. Functional Form, Limiting Reduction, and Critique

While MOND satisfies standard limiting reduction criteria (a smooth interpolating function $\mu$, regular limiting behavior as $a_0 \to 0$), modern appraisal has stressed that MOND’s reduction to Newtonian theory lacks theoretical grounding. The interpolating function $\mu(x)$ is inserted ad hoc, chosen for mathematical regularity and fit performance, not derived from underlying dynamics or a unified action principle (Antoniou et al., 2 Dec 2025). In the analysis by Palacios et al., two additional criteria—downward theoretical grounding (derivation from deeper principles) and upward grounding (natural embedding of Newtonian gravity)—highlight this deficit: there exists no fundamental theory (no “FUNDAMOND”) which selects $\mu(x)$, and Newtonian gravitation is not recoverable as a seamless sector within a single mathematical structure. Thus, MOND constitutes a “pathological reduction”: it achieves empirical success and formal limiting behavior but fails reduction-wise justification (Antoniou et al., 2 Dec 2025).

4. Physical Interpretations and Theoretical Embeddings

MOND’s mathematical structure admits multiple physical interpretations:

• Modified Gravity (AQUAL, QUMOND): Lagrangian-based modification of the Poisson equation via a nonlinear or quasi-linear operator acting on the potential, preserving conservation laws (Famaey et al., 28 Jan 2025, Famaey et al., 2011).
• Modified Inertia: Acceleration-dependent inertia, breaking Galilean invariance, possibly justified by violations of locality in low-acceleration regimes. Covariant implementations can generate the algebraic MOND law directly as a consequence of new symmetries among accelerated frames (Alzain, 2017, 1011.3618).
• Thermodynamic Origin: MOND as an emergent phenomenon on holographic screens, with interpolating function arising from quantum-statistical mechanics (e.g. Fermi energy sets $a_0$; crossover in specific heat yields functional form) (Pazy et al., 2011, Carranza et al., 2014).
• Extended Metric or Field Theories: By embedding Milgrom's scale in an f(R)-type metric theory, MOND's law emerges as a weak-field limit, breaking scale invariance and incorporating Noether symmetries connecting gravitational and MONDian sectors (Bernal et al., 2011).
• Particle-Based “Graviton Picture”: Gravity mediated by virtual massive gravitons can reproduce the MOND mass-discrepancy–acceleration relation with the “simple” μ(x) function, linking it to an explicit physical mechanism (Trippe, 2014, Trippe, 2013).
• Noncommutative Geometry: Smearing of matter in spacetime (D-brane scenario) mimics the deep-MOND regime, enforcing linear mass profiles and flat rotation curves without dark matter (Kuhfittig, 2022). These varied approaches reflect a lack of consensus on the underpinning microphysics, with most implementations requiring hand-crafted interpolating functions to reproduce the empirically observed phenomenology.

5. Observational Constraints, Failures, and Open Problems

MOND’s success is confined to rotationally supported galaxies; observed phenomena such as flat rotation curves and scaling relations are fitted without invoking dark matter (Desmond, 27 May 2025). However, systematic failures and tensions exist:

• Galaxy Clusters: MOND is unable to fully resolve mass discrepancies in clusters; lensing and thermodynamic profiles indicate a need for additional unseen mass or stronger gravitational force (Famaey et al., 28 Jan 2025, Desmond, 27 May 2025).
• Solar System and Binaries: External field effects and Solar System measurements (e.g., Cassini quadrupole constraints $Q_2$) impose bounds incompatible with the interpolating function required to fit galaxy-scale RAR data (Desmond, 27 May 2025, Iocco et al., 2015).
• Wide-Binary Tests: Empirical studies of wide binary stars have yielded conflicting results, with systematics complicating attempts to confirm or refute MONDian velocity boosts near $a_0$ (Desmond, 27 May 2025).
• Radial and Non-Radial Motions: MOND robustly predicts radial dynamics (RAR, BTFR), but struggles with non-radial and complex motion regimes, such as anisotropic dispersion or kinematic substructure.
• Cluster and Cosmological Scales: While some field-theoretic extensions can partially mitigate cluster discrepancy (e.g., inclusion of hot neutrinos), embedding MOND in a relativistic, cosmologically compatible theory remains incomplete (Famaey et al., 2011, Famaey et al., 28 Jan 2025).

6. Future Directions: Refinements and Fundamental Theory

Recent analyses advocate for improved reduction-wise justification for candidate modifications of gravity: a theory must link interpolating functions and transition regimes to dynamical microphysics or symmetries, embedding old limits non-arbitrarily within a coherent structure (Antoniou et al., 2 Dec 2025). Promising directions include:

• Screening and Frequency Scales: Limiting application of MONDian boosts to low-frequency (or slow) motions, as in time-nonlocal gravity or generalized inertia models (Desmond, 27 May 2025).
• Relativistic Embeddings: Formulation of TeVeS-like, Aether-Scalar-Tensor, or BIMOND models that recover both general relativity and MOND in appropriate limits, remain compatible with lensing and Solar System tests, and produce viable cosmologies (Famaey et al., 28 Jan 2025, Famaey et al., 2011).
• Empirical Probes: High-precision astrometry (Gaia DR4), weak-lensing surveys, wide-binary statistics, and resolved cluster X-ray and lensing maps are expected to decisively test MOND predictions in the next decade (Desmond, 27 May 2025).
• Emergent and Quantum Models: Statistical mechanics, entropic-force gravity, and noncommutative geometry are being explored for their capacity to derive MONDian regimes from first principles (Pazy et al., 2011, Carranza et al., 2014, Kuhfittig, 2022).

7. Summary Table: MOND Criteria and Status

Criterion (Editor’s term)	MOND Status	Reference
Empirical Phenomenology	Fits galactic rotation curves, RAR, BTFR	(Famaey et al., 28 Jan 2025)
Limiting Reduction	Formally reduces to Newtonian gravity	(Antoniou et al., 2 Dec 2025)
Theoretical Grounding	Absent: interpolating function ad hoc	(Antoniou et al., 2 Dec 2025)
Upward Grounding	Lacks unified structure embedding Newtonian	(Antoniou et al., 2 Dec 2025)
Relativistic Embedding	Partial; no universally accepted theory	(Famaey et al., 28 Jan 2025)
Cluster-Scale Discrepancy	Fails: needs extra dark mass or force	(Desmond, 27 May 2025)
Solar System Constraints	IF transition incompatible with SS data	(Desmond, 27 May 2025)
Emergent/Statistical Models	Under active investigation (holography, etc.)	(Pazy et al., 2011)

MOND remains a phenomenologically successful rule for galactic dynamics, offering striking regularities and tight scaling relations teased from observed baryons, but its reduction–wise justification and embedding in fundamental physics are not established. Progress requires a theory which naturally selects the empirical interpolation law, incorporates Newtonian gravity seamlessly, and passes all astrophysical and cosmological tests without arbitrary tuning.