Low-Acceleration and Modified-Dynamics Frameworks

Low-acceleration modified dynamics begins with a remarkably economical scaling law, but a theory must say where that law comes from. This chapter moves from the isolated spherical limit to action-based nonlinear fields, then separates modified gravity from modified inertia and asks whether a relativistic completion can describe light, clusters, cosmology, and healthy propagation. Its conclusion is a specification and consistency test, not an empirical verdict.

Helpful background. Weak-Field Gravity provides Poisson theory and weak metric potentials; Einstein Equations provides covariant metric dynamics; Gravitational Actions and Boundary Terms provides variational and conservation methods; Formulations of General Relativity provides constraints, modes, and initial-value language; and Gravity Beyond General Relativity provides the common comparison axes used across alternative theories.

Enter this chapter

The pages are ordered to prevent a phenomenological relation from being mistaken for a complete dynamics.

Page	Central question	Result carried forward
The Low-Acceleration Scaling Law	What follows from a universal acceleration scale and the deep-limit symmetry?	The asymptotic relation $g\simeq\sqrt{a_0g_N}$, flat circular speeds, and the baryonic mass–speed scaling, with symmetry assumptions exposed.
AQUAL and Quasilinear Modified Gravity	How can the rule follow from a conservative field theory?	Nonlinear and two-Poisson action principles, boundary data, and an anisotropic external-field limit.
Relativistic Modified-Dynamics Theories	Which extra fields can connect slow dynamics, lensing, and cosmology?	A physical matter metric, mode tests, and a dated comparison of scalar–vector–tensor constructions.
Modified Inertia versus Modified Gravity	Is changing the trajectory law equivalent to changing the sourced field?	A sharp action-level distinction and history-sensitive discriminants beyond circular motion.
Lensing, Clusters, and Cosmological Structure	Can the galaxy limit be exported to photons and the early universe?	A two-potential lensing test, residual cluster requirements, and the need to evolve extra cosmological modes.
Consistency Requirements and Observational Tests	What must be fixed before data can appraise a theory?	A theory specification, ordered consistency tests, agreement across scales, and predictions suitable for observational tests.

The relativistic page precedes the inertia comparison so that “modified dynamics” is not tacitly equated with one covariant theory. The final two pages then test whether the same specified construction survives changes of scale and observable.

The scaling core and its assumptions

Introduce an interpolation function with

$$\mu(g/a_0)g=g_N, \qquad \mu(x)\longrightarrow \begin{cases} 1, & x\gg1,\\ x, & x\ll1. \end{cases}$$

For a circular orbit outside an isolated spherical mass $M$, $g=V^2/r$ and $g_N=GM/r^2$. In the deep regime,

$$\frac{g^2}{a_0}=g_N \quad\Longrightarrow\quad V^4=GMa_0.$$

This is a dimension and normalization check: $GMa_0$ has units of speed to the fourth power. It also reveals the hidden assumptions. Spherical or one-dimensional symmetry is what makes the vector field algebraic; isolation fixes the asymptotic boundary condition; circular stationarity suppresses trajectory history. A disk in an external field, an eccentric orbit, or a cosmological perturbation does not inherit this equation unchanged. The scaling and its domain are reviewed in Famaey and McGaugh 2012, §§ 3–5.

From relation to dynamics

AQUAL changes the gravitational part of the action and yields

$$\nabla\cdot \left[ \mu\!\left(\frac{\lvert\nabla\Phi\rvert}{a_0}\right) \nabla\Phi \right]=4\pi G\rho.$$

Translation and rotation symmetry then support momentum and angular-momentum conservation. QUMOND performs two linear Poisson solves with a nonlinear effective source; it shares the spherical limit but is not the same theory in generic geometry. Both exhibit an external-field effect because the nonlinear operator retains the ambient field in a freely falling subsystem. The original action construction is in Bekenstein and Milgrom 1984, §§ II–III, and the quasilinear construction is in Milgrom 2010, §§ 2–5.

Modified inertia moves the change to a kinetic functional $S_k[\boldsymbol x(\cdot);a_0]$. Under the stated Galilei and limit requirements, a general realization is strongly nonlocal in time. Exact circular motion can obey the same algebraic law, while eccentric or multi-frequency trajectories need not. Thus the observationally convenient relation does not identify which side of the matter–field system was changed.

Relativity adds independent questions

With physical metric

$$d\tilde s^2=-(1+2\Psi)dt^2 +a^2(1-2\Phi)d\boldsymbol x^2,$$

slow motion mainly probes $\Psi$, whereas lensing probes $\Phi+\Psi$. A purely conformal scalar coupling preserves null trajectories and cannot by itself turn a dynamical enhancement into the corresponding enhanced light bending. TeVeS uses a dynamical vector to make the matter metric disformal; newer scalar–vector–tensor constructions organize the same requirements differently.

Every completion introduces new checks: count the extra modes after solving constraints, diagonalize their kinetic and gradient terms on the background used, derive the high-acceleration GR limit, state the strong-coupling scale, and evolve cosmological initial data. A successful quadratic calculation on Minkowski space is not a theorem for nonlinear cosmology or compact objects.

One ladder of regimes

A coherent assessment proceeds upward rather than transferring a formula by analogy:

1. derive the isolated symmetric scaling and its dimensional consequences;
2. solve the action-based field or trajectory equations with declared data;
3. take nonspherical, external-field, and many-body limits;
4. identify the physical metric and calculate both weak potentials;
5. evolve background and perturbations from early-universe initial data;
6. verify modes, hyperbolicity, cutoff control, compact objects, and radiation;
7. export fixed predictions and validity ranges for empirical appraisal.

At steps 4–6, additional fields or unseen matter may be essential. Such ingredients must remain in the theory’s stress-energy budget rather than being added only when a particular observable is considered.

Choose a route

Derive a galaxy limit. Start with the scaling page, then use AQUAL/QUMOND to see exactly when the algebraic law follows and how boundary data create an external-field effect.

Compare mechanisms. Read the inertia comparison alongside the action-based field page. Focus on eccentric motion, frequency dependence, composite bodies, and what is held fixed at a trajectory crossing.

Build a covariant theory. Continue through the relativistic and lensing/cosmology pages. Specify the matter metric before calculating an observable and test the modes on every background used.

Confront data. Complete the final theory tuple first. Then pass the fixed observable map to Tests of Gravity, where likelihoods, systematics, inference, and bounds belong.

Review the chapter

A proposal states only $g=\sqrt{a_0g_N}$ below $a_0$ and claims to predict isolated galaxies, cluster lensing, and cosmological growth. Derive the justified spherical circular result, then list what is missing from the other two claims.

Solution

For an isolated spherical mass, insert $g_N=GM/r^2$ into the stated deep relation. A circular orbit has $g=V^2/r$, so $V^4/r^2=a_0GM/r^2$ and hence $V^4=GMa_0$; the radius cancels. This establishes only the symmetry-reduced speed–mass scaling.

Cluster lensing needs an action-based field solution for a nonspherical, environment-dependent baryon distribution, a physical relativistic metric, both potentials $\Phi$ and $\Psi$, and any additional gravitating component. Cosmological growth additionally needs a background solution, extra-field perturbation equations, constraint-compatible primordial data, and a controlled nonlinear regime. Both claims require stability, branch, and cutoff checks. The algebraic galaxy relation supplies none of these inputs, so it cannot decide either prediction by itself.

References

• Bekenstein, Jacob, and Mordehai Milgrom (1984), “Does the Missing Mass Problem Signal the Breakdown of Newtonian Gravity?” Astrophysical Journal 286, 7–14. DOI.
• Famaey, Benoît, and Stacy S. McGaugh (2012), “Modified Newtonian Dynamics (MOND): Observational Phenomenology and Relativistic Extensions,” Living Reviews in Relativity 15, 10. DOI. Open PDF.
• Milgrom, Mordehai (2010), “Quasi-linear Formulation of MOND,” Monthly Notices of the Royal Astronomical Society 403, 886–895. DOI. Open PDF.

Dated Research assessment. For open questions and recent results, follow the link below. Each assessment states the date through which it considers results and the limits of its conclusions.

• Modified Gravity and the Dark-Matter Alternative