Modified Inertia versus Modified Gravity

The same circular-orbit scaling can arise by changing the gravitational field or by changing a body’s inertial response. Those are not two wordings of one theory. Modified gravity assigns a field to the source and retains the usual local particle kinetic term; modified inertia changes the trajectory action, so the response can depend on the orbit’s history and frequency content. The distinction becomes physical away from exactly circular, stationary motion. In one sentence: modified inertia changes the kinetic action and equation of motion, whereas modified gravity changes the Poisson equation for the gravitational field.

Required background. Weak-Field Gravity supplies Poisson fields, test-particle motion, and the weak equivalence principle.

Helpful background. Einstein Equations supplies the separation between source equations and matter equations; Gravitational Actions and Boundary Terms supplies variational and Noether reasoning; Formulations of General Relativity supplies initial-data language; Gravity Beyond General Relativity supplies degrees-of-freedom and limiting checks; and Relativistic Matter and Compact Objects supplies many-body and strong-field tests beyond ideal circular orbits.

Two different places to modify an action

In a modified-gravity model such as AQUAL, a particle retains

$$S_{\rm p}=\int dt\left(\frac12m\lvert\dot{\boldsymbol x}\rvert^2 -m\Phi(\boldsymbol x,t)\right),$$

so its equation of motion is local,

$$\ddot{\boldsymbol x}(t)=-\nabla\Phi(\boldsymbol x(t),t).$$

The change occurs in the source-to-field map: the mass distribution and boundary data determine $\Phi$ through a nonlinear or quasilinear field equation. Once that field has been solved, two test bodies at the same event with the same velocity receive the same instantaneous acceleration.

Modified inertia instead keeps, at least at the nonrelativistic level, the Newtonian source equation $\nabla^2\Phi_N=4\pi G\rho$ and replaces the kinetic action by a functional

$$S_k[\boldsymbol x(\cdot);a_0].$$

Its variation defines a trajectory functional

$$\boldsymbol{\mathcal A}[\boldsymbol x(\cdot);t,a_0] =-\nabla\Phi_N(\boldsymbol x(t)).$$

The Newtonian limit requires $\boldsymbol{\mathcal A}\to\ddot{\boldsymbol x}$ as $a_0\to0$. Galilei invariance, the Newtonian limit, and the deep low-acceleration scaling make a generic action of this kind strongly nonlocal in time; this is not equivalent to inserting a finite number of higher time derivatives. The argument and explicit action class are given in Milgrom 1994, §§ III and V.

Why circular orbits hide the distinction

The nonlocal trajectory functional above does not yet specify an initial-value problem. In particular, an ordinary single-history action with a bilocal kernel generally produces a time-symmetric variational equation. A causal retarded response needs a compatible effective description or an explicit evolution prescription. The kernel-variation calculation shows why choosing a retarded symbol in an action alone is insufficient.

For a stationary axisymmetric potential and an exactly circular orbit, symmetry collapses the trajectory functional to one radial scalar relation:

$$\mu\!\left(\frac{V^2}{Ra_0}\right)\frac{V^2}{R} =g_N(R), \qquad g_N(R)=\frac{d\Phi_N}{dR}.$$

This is the same algebraic form obtained from spherical modified gravity. It does not follow that the theories agree. An eccentric orbit contains a range of frequencies and accelerations; a vertical oscillation through a disk is not a circular radial mode; and a suddenly changed external field probes a memory kernel. In modified inertia, the response at time $t$ can depend on the entire relevant trajectory. In local modified gravity, history enters only through the evolved gravitational fields and their initial data.

The circular solution and its virial relation are derived in Milgrom 1994, §§ IV–V. The exact circular formula is therefore a symmetry-reduced prediction, not a universal pointwise replacement for $F=ma$.

Comparison by physical axis

Axis	Modified gravity	Modified inertia
Object changed	Source-to-field equation	Kinetic trajectory functional
Locality	Often local in space and time after fields are included	Generically nonlocal in time under the stated symmetry and limit requirements
Circular stationary orbit	Can reduce to an algebraic acceleration law	Can reduce to the same algebraic law
Noncircular motion	Determined by the local modified field	Can depend on frequencies, eccentricity, and past trajectory
External-field effect	Arises from nonlinearity of the field equation and its boundary data	Arises through the trajectory functional and may depend on internal/external frequency ratios
Many-body system	Requires solving one field sourced by all bodies	Requires a consistent composite-body action; a test-particle rule alone is insufficient
Relativistic completion	Must specify a physical matter metric and extra field modes	Must explain covariant causal memory or additional degrees of freedom as well as light propagation

Both routes can conserve energy and angular momentum when their actions have time-translation and rotation symmetry. In a nonlocal theory, however, the Noether charges need not be the instantaneous Newtonian expressions. Writing an algebraic acceleration rule without an action is insufficient to establish these conservation laws in either route.

A clean discriminant

Imagine two probes that meet at the same location with the same instantaneous position, velocity, and Newtonian force, but arrived through trajectories with different frequency histories. A local modified-gravity field predicts the same acceleration at that event once the field state is fixed. A genuinely nonlocal modified-inertia functional may predict different responses. This is a conceptual discriminant; a quantitative test requires a specified kernel, initial-history prescription, and treatment of composite bodies.

The phenomenological distinction is reviewed in Famaey and McGaugh 2012, §§ 3.2 and 6.3. It also prevents an empirical fit of the circular relation from deciding which mechanism, if either, is responsible.

Common pitfalls

Replacing acceleration algebraically defines modified inertia. It defines only a special force law. A viable modified-inertia theory needs an action or equivalent evolution prescription, conserved quantities, and initial-history data.

Nonlocality automatically means acausality. A time-nonlocal effective action does not by itself specify whether the physical evolution uses retarded, advanced, or mixed data. Causality can be judged only after the kernel and its relativistic completion are stated.

Exercise

Two bodies cross the same point with equal velocity in a static Newtonian potential. One has spent many periods on a nearly circular orbit; the other has just arrived on a highly eccentric orbit. Which framework can distinguish their instantaneous accelerations, and what assumptions are needed?

Solution

For a local modified-gravity theory with a fixed, already-solved field, the equation $\ddot{\boldsymbol x}=-\nabla\Phi$ assigns the same acceleration to both bodies at the crossing event. Their different histories do not enter the test-particle equation. A nonlocal modified-inertia functional can distinguish them because its value at the event may depend on the past spectrum and shape of each trajectory. That conclusion requires a specified universal kinetic functional and compatible history data; the circular algebraic formula alone cannot calculate the difference. If the gravitational field itself is time-dependent, its state must also be held fixed before attributing a difference to inertia rather than to field evolution.

References

• 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 (1994), “Dynamics with a Nonstandard Inertia–Acceleration Relation: An Alternative to Dark Matter,” Annals of Physics 229, 384–415. DOI. Open PDF.