The Low-Acceleration Scaling Law

The low-acceleration scaling law relates the actual centripetal acceleration to the Newtonian field of the baryons through one acceleration scale $a_0$. In its deep regime, $g^2=a_0g_N$. For an isolated finite mass this turns the Newtonian $r^{-2}$ field into an $r^{-1}$ field, producing an asymptotically constant circular speed and the mass–speed relation $V_\infty^4=GMa_0$. Those consequences are kinematic; a complete theory must still specify whether gravity or inertia is modified.

Required background. Weak-Field Gravity supplies the Newtonian potential, Poisson equation, and orbital limit.

Helpful background. Einstein Equations supplies the relativistic completion problem; Gravitational Actions and Boundary Terms supplies conservation laws derived from an action; Formulations of General Relativity supplies initial and boundary data; Gravity Beyond General Relativity supplies the distinction between phenomenological scaling and a complete theory; Relativistic Matter and Compact Objects supplies many-body and strong-field regimes where the simple law is insufficient.

The interpolation law

Let baryonic matter of density $\rho_b$ produce the Newtonian potential

$$\nabla^2\Phi_N=4\pi G\rho_b, \qquad \boldsymbol g_N=-\boldsymbol\nabla\Phi_N.$$

In spherical, planar, or cylindrical one-dimensional symmetry, the low-acceleration relation can be written in either form

$$\mu\!\left(\frac{g}{a_0}\right)\boldsymbol g =\boldsymbol g_N,$$ $$\boldsymbol g =\nu\!\left(\frac{g_N}{a_0}\right)\boldsymbol g_N.$$

The two interpolation functions are inverse descriptions only after their arguments are related by

$$y=x\mu(x), \qquad \nu(y)=\frac{1}{\mu(x)}.$$

The required asymptotic limits are

$$\mu(x)\longrightarrow1 \quad(x\gg1), \qquad \mu(x)\longrightarrow x \quad(x\ll1),$$

or equivalently

$$\nu(y)\longrightarrow1 \quad(y\gg1), \qquad \nu(y)\longrightarrow y^{-1/2} \quad(y\ll1).$$

Thus high acceleration recovers Newtonian gravity, whereas the deep regime gives

$$\boxed{g=\sqrt{a_0g_N}}.$$

The relation, its interpolation conventions, and its restricted exactness in one-dimensional symmetry are reviewed in Famaey and McGaugh 2012, §§ 3.1–3.3. For a generic nonsymmetric source, the algebraic vector relation need not be curl free and is not by itself a complete gravitational field equation.

Flat rotation from a finite baryonic mass

Outside an isolated spherical source of baryonic mass $M$,

$$g_N(r)=\frac{GM}{r^2}.$$

The transition occurs near the MOND radius

$$r_M=\sqrt{\frac{GM}{a_0}},$$

where $g_N=a_0$. For $r\gg r_M$,

$$g(r) =\sqrt{a_0\frac{GM}{r^2}} =\frac{\sqrt{GMa_0}}{r}.$$

A circular orbit satisfies $V_c^2/r=g$, so

$$V_c^2=\sqrt{GMa_0}, \qquad \boxed{V_\infty^4=GMa_0}.$$

The speed is independent of radius, and the fourth power of the asymptotic speed is proportional to baryonic mass. The relation follows without choosing an interpolation function because only the deep asymptote was used.

Dimensional analysis is a strong check: $[GMa_0]=L^4T^{-4}=[V^4]$. The same balance defines a characteristic surface density of order $a_0/G$; numerical factors depend on whether one uses a spherical mean, a disk, or a particular convention.

Potential and boundary behavior

Here $g=\lvert\boldsymbol g\rvert=d\Phi/dr\gt0$ is the inward acceleration magnitude, whereas $\boldsymbol g=-g\hat{\boldsymbol r}$. In a modified-gravity realization, the deep isolated field integrates to

$$\Phi(r) =\sqrt{GMa_0}\ln\!\left(\frac{r}{r_0}\right).$$

Unlike the Newtonian potential, this logarithm does not approach a finite constant at infinity. The reference scale $r_0$ is irrelevant to local forces, but the asymptotic behavior shows that an exactly isolated deep-regime system cannot always be treated with the usual boundary condition $\Phi\to0$. Embedding in an external field or a cosmological environment changes the large-radius boundary problem.

This is one reason the scaling law is not a full theory. An action-based field equation must determine nonspherical curl fields, conservation laws, many-body forces, and the external-field effect. A modified-inertia theory instead changes the trajectory functional and can reproduce the circular relation while predicting different behavior for noncircular histories.

Interpolation is model information

The asymptotic limits do not fix the transition near $g\sim a_0$. For example, many monotone functions satisfy the two limits but differ in how rapidly they approach Newtonian behavior. A chosen $\mu$ or $\nu$ is therefore part of a model, along with the field equation or kinetic action in which it appears.

Uniqueness also matters. In QUMOND, a sufficient one-dimensional monotonicity condition is

$$\nu(y)+y\nu'(y)\gt0.$$

It ensures that $y\nu(y)$ increases rather than folding into multiple algebraic branches. Milgrom derives the QUMOND relation, deep limit, and uniqueness condition in Milgrom 2010, § 2, equations (3)–(9).

The Newtonian limit may be expressed as $a_0\to0$ at fixed source and finite radius, for which $g_N/a_0\to\infty$ and $\nu\to1$. It is nonuniform if the radius is simultaneously taken beyond $r_M\propto a_0^{-1/2}$. The order of limits must be stated.

Common pitfalls

Apply the algebraic vector law to every geometry. Away from one-dimensional symmetry, $\nu\boldsymbol g_N$ generally has nonzero curl. An action-based field equation supplies the required gradient projection.

Treat $a_0$ as a force. It is an acceleration scale. The force also depends on test mass, and the transition radius depends on source mass.

Infer a relativistic lensing law from circular speeds. Lensing depends on the physical spacetime metric and its two weak-field potentials, which the nonrelativistic scaling does not specify.

Exercises

For an isolated spherical baryonic mass $M$, derive the transition radius, deep-regime field, potential, and asymptotic circular speed. Then explain why the limit $a_0\to0$ is not uniform over all radii.

Solution

Set $g_N=GM/r^2=a_0$ to obtain

$$r_M=\sqrt{\frac{GM}{a_0}}.$$

For $r\gg r_M$, the deep relation gives

$$g=\sqrt{a_0g_N} =\frac{\sqrt{GMa_0}}{r}.$$

Integrating $d\Phi/dr=g$, with $\boldsymbol g=-g\hat{\boldsymbol r}$, gives

$$\Phi(r)=\sqrt{GMa_0}\ln(r/r_0).$$

Circular balance yields

$$\frac{V_c^2}{r}=g \quad\Longrightarrow\quad V_c^4=GMa_0.$$

At any fixed finite $r$, taking $a_0\to0$ makes $g_N/a_0\to\infty$, so $\nu\to1$ and $g\to g_N$. But $r_M\propto a_0^{-1/2}$ moves outward. If $r$ is scaled with or beyond $r_M$, the system remains in the transition or deep regime. Hence the Newtonian limit is pointwise at fixed radius, not uniform on an unbounded domain.

References

• Famaey, Benoit, 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.