AQUAL and Quasilinear Modified Gravity

The algebraic low-acceleration rule is exact only in special symmetries. AQUAL replaces it with a nonlinear elliptic field equation derived from an action; QUMOND uses two linear Poisson solves separated by one nonlinear source construction. Both preserve total momentum and angular momentum, yet their internal dynamics depends on a uniform external gravitational field. That external-field effect violates the strong equivalence principle without making test-body free fall composition dependent.

Required background. Weak-Field Gravity supplies Poisson theory, potentials, and weak-equivalence-principle language.

Helpful background. Einstein Equations supplies the distinction between weak and strong equivalence; Gravitational Actions and Boundary Terms supplies Noether conservation and elliptic boundary terms; Formulations of General Relativity supplies constraint and boundary-value methods; Gravity Beyond General Relativity supplies branch and limiting checks; Relativistic Matter and Compact Objects supplies the many-body systems for which an algebraic test-particle rule fails.

AQUAL from an action

Let matter of density $\rho$ move with standard Newtonian inertia in a potential $\Phi$. The nonrelativistic action is

$$S=\int dt\,d^3x \left[ \frac12\rho v^2-\rho\Phi -\frac{a_0^2}{8\pi G} \mathcal F\!\left(\frac{\lvert\nabla\Phi\rvert^2}{a_0^2}\right) \right].$$

Write

$$z=\frac{\lvert\nabla\Phi\rvert^2}{a_0^2}, \qquad \mu(\sqrt z)=\mathcal F'(z).$$

Varying $\Phi$ and integrating by parts gives

$$\delta S =\int dt\,d^3x \left[ \frac{1}{4\pi G} \nabla\cdot\left(\mu\nabla\Phi\right)-\rho \right]\delta\Phi,$$

so the AQUAL field equation is

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

The boundary term vanishes only for the declared asymptotic or finite-domain data. Spatial translations and rotations of the action yield conserved total momentum and angular momentum. This is a decisive improvement over applying $\boldsymbol g=\nu\boldsymbol g_N$ point by point in a generic geometry, where the right-hand side can have nonzero curl. The AQUAL construction and its conservation laws originate in Bekenstein and Milgrom 1984, § II, equations (2)–(9).

For spherical symmetry, integrate the equation over a sphere:

$$4\pi r^2\mu(g/a_0)g=4\pi GM(r).$$

Therefore

$$\mu(g/a_0)g=g_N,$$

and the algebraic scaling law is recovered. This Gauss-law calculation is an independent normalization and sign check.

Calling the equation elliptic requires a condition on the interpolation function. Linearization about a background with nonzero gradient gives the spatial matrix $\mu\,\delta_{ij}+x\mu'(x)e_i e_j$, where $x=\lvert\nabla\Phi\rvert/a_0$ and $\boldsymbol e$ points along that gradient. Its eigenvalues are $\mu$ twice and $\mu+x\mu'$ once, so require $\mu\gt0$ and $\mu+x\mu'\gt0$. The deep limit $\mu\sim x$ is degenerately elliptic at a zero of the field; uniform-ellipticity claims cannot be extended through such a point without further analysis.

QUMOND as a projected algebraic field

QUMOND introduces the Newtonian potential $\Phi_N$. Its two-potential gravitational and interaction action is

$$S=\int dt\,d^3x\left[ -\rho\Phi-\frac{1}{8\pi G} \left\{2\nabla\Phi\cdot\nabla\Phi_N -a_0^2\mathcal Q\!\left(\frac{\lvert\nabla\Phi_N\rvert^2}{a_0^2}\right) \right\}\right],$$

with the standard matter kinetic term understood and $\mathcal Q'(z)=\nu(\sqrt z)$. Varying $\Phi$ first gives

$$\nabla^2\Phi_N=4\pi G\rho,$$

while varying $\Phi_N$ gives

$$\nabla^2\Phi =\nabla\cdot\left[ \nu\!\left(\frac{\lvert\nabla\Phi_N\rvert}{a_0}\right) \nabla\Phi_N \right].$$

Thus the algorithm is:

1. solve one linear Poisson equation for $\Phi_N$;

2. evaluate the nonlinear effective source

   $$\widehat\rho =\frac{1}{4\pi G} \nabla\cdot\left(\nu\nabla\Phi_N\right);$$

3. solve a second linear Poisson equation for $\Phi$.

The vector $\nu\nabla\Phi_N$ need not itself be a gradient. The second Poisson solve projects it onto the gradient sector compatible with the boundary data. Because QUMOND also follows from a two-potential action, the projection retains the standard translation and rotation conservation laws. Its action, field equations, and gradient decomposition are derived in Milgrom 2010, § 2, equations (3)–(9).

AQUAL and QUMOND coincide with the algebraic rule in one-dimensional symmetry, but they need not agree for a disk, interacting galaxies, or an external field. They are distinct modified-gravity theories sharing the same asymptotic scaling.

The external-field effect

Suppose a small subsystem has internal potential $\phi$ and lies in a nearly uniform external field $\boldsymbol g_e=g_e\boldsymbol e$. Write

$$\Phi=-\boldsymbol g_e\cdot\boldsymbol x+\phi, \qquad \lvert\nabla\phi\rvert\ll g_e.$$

Define

$$\mu_e=\mu(g_e/a_0), \qquad L_e= \left.\frac{d\ln\mu}{d\ln x}\right|_{x=g_e/a_0}.$$

Linearizing the AQUAL flux gives

$$\mu_e \left( \delta_{ij}+L_e e_i e_j \right)\partial_j\phi.$$

The internal equation becomes

$$\boxed{ \mu_e\left( \nabla^2\phi+L_e\partial_\parallel^2\phi \right)=4\pi G\rho }.$$

Even though the external acceleration is spatially uniform, it changes the effective internal coupling and makes the response anisotropic. A freely falling coordinate transformation removes the uniform potential from the particle equation but not from the nonlinear field operator. This is the external-field effect.

Test particles still follow $\ddot{\boldsymbol x}=-\nabla\Phi$ independently of composition, so the weak equivalence principle survives in this nonrelativistic model. The strong equivalence principle fails because local self-gravitating physics retains information about the ambient gravitational field. The AQUAL and QUMOND external-field regimes are reviewed in Famaey and McGaugh 2012, §§ 6.1–6.3.

Boundary data and limiting regimes

For an isolated deep-regime mass, $\Phi$ grows logarithmically, so the usual condition $\Phi\to0$ at spatial infinity is inappropriate. In an external field, the far solution crosses to the anisotropic quasi-Newtonian equation above. Numerical domains must therefore encode the ambient field rather than truncate an isolated logarithm arbitrarily.

Three limits check the equations:

• $g_e/a_0\gg1$ gives $\mu_e\to1$ and $L_e\to0$, recovering Poisson theory;
• $g_e/a_0\ll1$ with internal field still smaller gives $\mu_e\simeq g_e/a_0$ and $L_e\simeq1$ in the deep asymptote;
• spherical isolation reduces both AQUAL and QUMOND to $\mu(g/a_0)g=g_N$.

None of these nonrelativistic limits determines light deflection or cosmological perturbations. Those require a relativistic matter metric.

Common pitfalls

Momentum conservation follows from the algebraic rule. It need not. It follows from the translation-invariant AQUAL or QUMOND action.

A uniform external field has no local effect. That is true in Newtonian gravity for a freely falling subsystem, but false for a nonlinear low-acceleration field equation.

AQUAL and QUMOND are the same away from symmetry. They share asymptotes and special symmetric solutions, not their complete nonsymmetric fields.

Exercises

Starting from the AQUAL action, derive the field equation and its linearized form for a subsystem dominated by a uniform external field. Identify the Newtonian limit and the source of anisotropy.

Solution

Variation of the field term gives

$$\delta S_{\Phi} =-\frac{1}{4\pi G}\int \mu\,\nabla\Phi\cdot\nabla\delta\Phi\,dtd^3x.$$

After integration by parts and addition of $-\int\rho\delta\Phi\,dtd^3x$, stationarity yields

$$\nabla\cdot(\mu\nabla\Phi)=4\pi G\rho.$$

Set $\nabla\Phi=-g_e\boldsymbol e+\nabla\phi$ and expand to first order in $\nabla\phi/g_e$. The derivative of $\mu$ contributes only along $\boldsymbol e$, giving the response tensor

$$\mu_e(\delta_{ij}+L_e e_i e_j).$$

Taking its divergence produces

$$\mu_e(\nabla^2\phi+L_e\partial_\parallel^2\phi) =4\pi G\rho.$$

For $g_e/a_0\gg1$, $\mu_e\to1$ and $L_e\to0$, so the ordinary Poisson equation is recovered. For finite $L_e$, derivatives parallel and perpendicular to the external field have different coefficients. The anisotropy comes from the field dependence of $\mu$, not from a tidal gradient of the external potential.

References

• Bekenstein, Jacob D., and Mordehai Milgrom (1984), “Does the Missing Mass Problem Signal the Breakdown of Newtonian Gravity?” The Astrophysical Journal 286, 7–14. DOI.
• 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.