Relativistic MOND (RMOND): Covariant Extensions

• Relativistic MOND (RMOND) is a class of covariant gravitational theories that extend Modified Newtonian Dynamics to recover general relativity in high-acceleration regimes while exhibiting scale-invariant MOND behavior at low accelerations.
• These theories employ diverse frameworks—including metric extensions, scalar-vector-tensor models, preferred foliations, and nonlocal formulations—to address galaxy rotation curves, gravitational lensing, and cosmological observations.
• RMOND proposals offer practical insights by predicting flat galactic rotation curves, dark matter–mimicking lensing effects, and testable cosmological scenarios that connect the MOND acceleration constant to cosmic parameters.

Relativistic MOND (RMOND, also written RelMOND in some constructions) denotes covariant completions of Modified Newtonian Dynamics that aim to recover general relativity (GR) in the high-acceleration regime while reproducing MOND phenomenology in the low-acceleration regime, where dynamics become scale invariant and are governed by an acceleration constant $a_0$. In the literature, RMOND is not a single theory but a class of proposals: modified-gravity, modified-inertia, scalar-vector-tensor, preferred-foliation, nonlocal metric, gauge-theoretic, brane-world, and thermodynamic constructions all appear under this heading. A recurring theme is that any viable RMOND must address not only galaxy dynamics but also gravitational lensing, cosmology, and, in modern formulations, gravitational-wave constraints (Milgrom, 2014, Skordis et al., 2020, Finster et al., 2023).

1. Defining principles and deep-MOND structure

The common starting point is the MOND postulate that standard dynamics is recovered for accelerations much larger than $a_0$, while the low-acceleration regime $g\ll a_0$ approaches a deep-MOND limit that is scale invariant under

$(t,\mathbf{r}) \to \lambda (t,\mathbf{r}).$

In this limit, the combination

$A_0 \equiv G a_0$

replaces $G$ and $a_0$ as the relevant constant. The standard nonrelativistic modified-Poisson form used as the target weak-field limit is

$$\nabla\cdot\left[\mu\!\left(\frac{|\nabla\phi|}{a_0}\right)\nabla\phi\right] = 4\pi G\rho,$$

with $\mu(x)\to 1$ for $x\to\infty$ and $a_0$0 for $a_0$1. In the deep-MOND regime, this yields the asymptotic baryonic relation

$a_0$2

and the asymptotic lensing angle scales as

$a_0$3

These are treated as defining phenomenological constraints on any relativistic completion (Milgrom, 2014).

A second foundational element is the cosmological coincidence

$a_0$4

which motivates attempts to derive $a_0$5 from de Sitter structure, the Hubble scale, or vacuum physics rather than inserting it purely phenomenologically. This coincidence underlies several later RMOND programs, including nonlocal metric models, right-handed gauge-sector constructions, brane-world pictures, and theories in which $a_0$6 may vary with cosmic time or emerge only in a nonrelativistic sector (Milgrom, 2014, Milgrom, 2018, Singh, 7 Jan 2026).

2. Metric and geometric completions

One major line of development realizes MOND as the weak-field limit of a relativistic metric theory. In extended metric gravity, Milgrom’s acceleration constant is promoted to a fundamental scale by defining

$a_0$7

and constructing the action

$a_0$8

For the power-law choice $a_0$9, the weak-field, low-acceleration limit gives

$g\ll a_0$0

so MOND arises in the same structural sense that Newtonian gravity arises from the weak-field limit of GR. In this framework, $g\ll a_0$1 becomes a fundamental constant of the gravitational theory and scale invariance is explicitly broken; the mass dependence is carried by $g\ll a_0$2, built from $g\ll a_0$3 and $g\ll a_0$4 (Bernal et al., 2011, Mendoza et al., 2012).

The same metric program was extended in several directions. A Noether-symmetry analysis of the $g\ll a_0$5 model identified a conserved quantity proportional to $g\ll a_0$6, reinforcing the interpretation that a relativistic MOND theory must encode both the Schwarzschild scale and the MOND scale (Bernal et al., 2011). A conformal-geometry construction introduced a local expansion factor $g\ll a_0$7, a modified connection $g\ll a_0$8, and a covariant acceleration potential

$g\ll a_0$9

with point-source solution

$(t,\mathbf{r}) \to \lambda (t,\mathbf{r}).$0

In that model the logarithmic term dominates at large radius and yields the $(t,\mathbf{r}) \to \lambda (t,\mathbf{r}).$1 acceleration law needed for flat rotation curves (Chadwick et al., 2013).

A more elaborate local construction added torsion and derivative matter couplings. In that theory the geometry is metric-affine with torsion tensor

$(t,\mathbf{r}) \to \lambda (t,\mathbf{r}).$2

and the final action combines an $(t,\mathbf{r}) \to \lambda (t,\mathbf{r}).$3 sector with terms involving derivatives of the matter Lagrangian. Specializing to $(t,\mathbf{r}) \to \lambda (t,\mathbf{r}).$4 and suitable derivative couplings yields the MONDian weak-field scaling

$(t,\mathbf{r}) \to \lambda (t,\mathbf{r}).$5

The paper is explicit that the resulting matter-Lagrangian dependence inside gravity is nonstandard and that the model is intended for systems with acceleration $(t,\mathbf{r}) \to \lambda (t,\mathbf{r}).$6, not as a universal theory at all scales (Barrientos et al., 2016).

3. Preferred foliation, scalar-vector-tensor theories, and khronometric RMOND

A second major branch uses additional gravitational fields, especially timelike vectors or preferred foliations. Milgrom’s review identifies TeVeS as the first successful relativistic MOND theory; it employs a metric $(t,\mathbf{r}) \to \lambda (t,\mathbf{r}).$7, vector field $(t,\mathbf{r}) \to \lambda (t,\mathbf{r}).$8, scalar field $(t,\mathbf{r}) \to \lambda (t,\mathbf{r}).$9, and a physical metric

$A_0 \equiv G a_0$0

The same review also discusses Einstein-Aether MOND adaptations, with a unit timelike vector and Lagrangian $A_0 \equiv G a_0$1, and BIMOND, a bimetric theory in which the interaction is built from the difference of Levi-Civita connections of two metrics (Milgrom, 2014).

The khronon formulation specializes the preferred-frame idea to a scalar field $A_0 \equiv G a_0$2 whose level sets define a preferred time foliation. The unit normal is

$A_0 \equiv G a_0$3

and the MOND modification is encoded in the acceleration of the normal congruence,

$A_0 \equiv G a_0$4

through the action

$A_0 \equiv G a_0$5

This theory can be written either covariantly in four dimensions or in a $A_0 \equiv G a_0$6 form with $A_0 \equiv G a_0$7; the two formulations are explicitly shown to be equivalent. In the nonrelativistic limit, the field equations reduce to

$A_0 \equiv G a_0$8

so the MOND interpolation function is identified as $A_0 \equiv G a_0$9 (Blanchet et al., 2012).

The Blanchet–Marsat theory was reanalyzed in khronometric language to test whether the slow-motion limit is consistent and whether the stationary MOND solutions are stable. The theory uses the action

$G$0

and the analysis shows that stationary MOND solutions are recovered in the slow-motion limit and are stable in the deep-MOND regime for spherical, cylindrical, and planar symmetry, provided

$G$1

with $G$2. The same analysis also finds that for nonstationary systems in the low-acceleration regime the khronon field generally gives an order-unity correction to MOND, so the relativistic theory is not merely MOND plus small relativistic corrections (Flanagan, 2023).

The Skordis–Zlosnik theory is a newer scalar-vector-tensor realization, built from the spacetime metric $G$3, a scalar field $G$4, and a unit timelike vector field $G$5. Its action contains the Einstein term plus a MOND sector involving

$G$6

and was designed to recover MOND locally while fitting the cosmic microwave background and linear matter power spectrum. Its quadratic action is reported to be free of ghost instabilities, the weak-field theory has $G$7, and tensor modes propagate at the speed of light (Skordis et al., 2020).

Subsequent analyses have developed the cosmology of this class. Phase-space analysis finds a viable sequence of radiation, matter, and de Sitter epochs for several choices of $G$8, while ruling out the simplest quadratic potential as cosmologically nonviable (Kashfi et al., 2022). Relaxing the usual static background assumption allows time-dependent effective $G$9 and time-dependent $a_0$0, with specific choices of $a_0$1 reproducing the redshift dependence of $a_0$2 reported from the Magneticum cold dark matter simulations; the same work reconfirms that the theory has only two tensor polarizations and luminal tensor speed (Tian et al., 2023). A further perturbative study derives the post-Newtonian and exact relativistic perturbation equations, shows that baryon perturbations grow faster in the MOND regime, interprets the MOND field as a fluid with specific equation of state and no anisotropic stress, and derives a Jeans criterion for the MOND field (Hwang et al., 2024).

A recent reformulation connects this entire scalar-vector-tensor sector to mimetic gravity. In that framework, any relativistic MOND theory with a unit-timelike vector field, including TeVeS and AeST, can be embedded in a conformal/disformal-invariant parent theory. Gauge-fixing can impose $a_0$3, $a_0$4, or $a_0$5, so the usual vector normalization constraint, the mimetic scalar constraint, and the cross-contraction constraint become interchangeable as long as the vector and scalar remain timelike (Domènech et al., 14 Mar 2025).

4. Alternative mechanisms and unification programs

RMOND has also been pursued through mechanisms that do not fit neatly into standard modified-gravity or scalar-vector-tensor templates.

Approach	Core ingredients	Stated MOND mechanism
Dark electromagnetism	$a_0$6; charge $a_0$7	Coulomb-like force with $a_0$8 gives effective $a_0$9 falloff
Modified energetics	Conserved $$\nabla\cdot\left[\mu\!\left(\frac{|\nabla\phi|}{a_0}\right)\nabla\phi\right] = 4\pi G\rho,$$0, second metric $$\nabla\cdot\left[\mu\!\left(\frac{|\nabla\phi|}{a_0}\right)\nabla\phi\right] = 4\pi G\rho,$$1	MOND enters the matter energetics rather than the Einstein tensor
Nonlocal metric MOND	Inverse d’Alembertian, nonlocal scalars, retarded boundary conditions	Tully–Fisher and weak lensing from nonlocal metric field equations
Brane-world MOND	Nearly spherical brane, external potential $$\nabla\cdot\left[\mu\!\left(\frac{|\nabla\phi|}{a_0}\right)\nabla\phi\right] = 4\pi G\rho,$$2	$$\nabla\cdot\left[\mu\!\left(\frac{|\nabla\phi|}{a_0}\right)\nabla\phi\right] = 4\pi G\rho,$$3 from global brane balance
Entropic RMOND	Debye-corrected equipartition and Unruh temperature	Thermal correction $$\nabla\cdot\left[\mu\!\left(\frac{|\nabla\phi|}{a_0}\right)\nabla\phi\right] = 4\pi G\rho,$$4 modifies Einstein equations in low-acceleration regime
Minimal IR metric deformation	Right-handed gauge sector and $$\nabla\cdot\left[\mu\!\left(\frac{|\nabla\phi|}{a_0}\right)\nabla\phi\right] = 4\pi G\rho,$$5	UV-vanishing metric deformation yields AQUAL in the deep-MOND limit

In the dark-electromagnetism proposal, the central claim is that GR and RelMOND are analogues of broken electroweak symmetry, with

$$\nabla\cdot\left[\mu\!\left(\frac{|\nabla\phi|}{a_0}\right)\nabla\phi\right] = 4\pi G\rho,$$6

GR is identified with the broken $$\nabla\cdot\left[\mu\!\left(\frac{|\nabla\phi|}{a_0}\right)\nabla\phi\right] = 4\pi G\rho,$$7 sector, while RelMOND is identified with the unbroken $$\nabla\cdot\left[\mu\!\left(\frac{|\nabla\phi|}{a_0}\right)\nabla\phi\right] = 4\pi G\rho,$$8. The source charge is taken to be

$$\nabla\cdot\left[\mu\!\left(\frac{|\nabla\phi|}{a_0}\right)\nabla\phi\right] = 4\pi G\rho,$$9

and the deep-MOND scaling is recovered by introducing the effective distance

$\mu(x)\to 1$0

Because $\mu(x)\to 1$1 is treated as a fixed cosmological scale, the resulting force law falls effectively as $\mu(x)\to 1$2, reproducing the deep-MOND form $\mu(x)\to 1$3. The same paper argues that DEM can mimic cold dark matter in CMB anisotropies, gravitational lensing, and large-scale structure phenomenology (Finster et al., 2023).

A different proposal places MOND in the matter sector rather than the curvature sector. In “modified energetics,” Einstein’s equation retains its usual left-hand side but the source becomes

$\mu(x)\to 1$4

with $\mu(x)\to 1$5 a second metric generated through a gravitational Higgs-like mechanism. The model is explicitly presented as a relativistic extension of modified inertia, and the authors emphasize that no explicit action principle is provided (Demir et al., 2014).

Nonlocal metric MOND instead keeps matter coupled normally to the metric and attributes MOND to nonlocal corrections in the gravitational field equations,

$\mu(x)\to 1$6

In that program, nonlocal scalars built from inverse d’Alembertians are motivated by vacuum polarization of infrared gravitons produced during primordial inflation. The models are designed to reproduce the Tully–Fisher relation together with sufficient weak lensing, and the cosmological branch of the nonlocal function is left available for reconstruction (Woodard, 2014).

The brane-world picture is heuristic rather than a complete relativistic theory, but it gives a geometrical interpretation of the MOND scale. The universe is modeled as a nearly spherical brane with

$\mu(x)\to 1$7

and the acceleration scale is identified as

$\mu(x)\to 1$8

The model explicitly distinguishes a nonrelativistic regime $\mu(x)\to 1$9 from a relativistic regime $x\to\infty$0, and suggests that $x\to\infty$1 may become potential-dependent or lose its status altogether in the relativistic domain (Milgrom, 2018).

Recent work has extended RMOND into thermodynamic and gauge-unification settings. A modified entropic-gravity theory introduces a Debye-like correction to equipartition,

$x\to\infty$2

and derives modified Einstein equations with explicit thermal corrections; in the low-temperature, weak-field regime the solution yields

$x\to\infty$3

and a fit to NGC 3198 gives $x\to\infty$4, compared with $x\to\infty$5 and $x\to\infty$6, with improved agreement at $x\to\infty$7 (Rostami et al., 7 Nov 2025). A separate 2026 proposal derives gravity from an $x\to\infty$8 connection via a Plebanski/MacDowell–Mansouri mechanism and implements MOND by a UV-vanishing infrared metric deformation,

$x\to\infty$9

with the deep-MOND limit selected by conformal symmetry in three spatial dimensions, whose group is isomorphic to $a_0$00, and with

$a_0$01

That construction states explicitly that GR is recovered exactly at high acceleration and the Bekenstein–Milgrom AQUAL equation emerges at low acceleration without introducing additional propagating fields beyond those already present in a right-handed gauge sector (Singh, 7 Jan 2026).

An earlier phenomenological FRW attempt introduced an “Inverse Yukawa Field” and replaced a constant cosmological term by a distance-dependent term $a_0$02 in

$a_0$03

with the stated goal of reproducing MOND-like galactic behavior, a modified critical density, and an FRW cosmology without nonbaryonic dark matter. This was presented as a heuristic bridge between MOND and relativistic cosmology rather than as a complete covariant field theory (Falcon, 2010).

5. Cosmology, lensing, perturbations, and gravitational waves

A defining distinction between MOND and RMOND is that the latter must handle relativistic observables. Milgrom’s review emphasizes that covariant MOND theories can be written with correct gravitational lensing, and several constructions make this a design principle rather than a derived afterthought. In the Skordis–Zlosnik theory the vector field is essential precisely because a purely conformal scalar-tensor theory would not give enough lensing; the weak-field limit has

$a_0$04

so once the potential mimics a dark-matter potential, lensing follows accordingly. The 2026 infrared-metric-deformation model similarly states that there is no gravitational slip in the quasistatic regime, again with $a_0$05 (Milgrom, 2014, Skordis et al., 2020, Singh, 7 Jan 2026).

Cosmology has become the main discriminant among RMOND proposals. The Skordis–Zlosnik program was introduced explicitly to fit the observed cosmic microwave background and linear matter power spectrum without particle dark matter (Skordis et al., 2020). Its phase-space analysis shows a viable radiation $a_0$06 matter $a_0$07 de Sitter sequence for Higgs-like, cosh, and exponential choices of $a_0$08, while the simplest quadratic choice is not viable (Kashfi et al., 2022). Further work demonstrates that, once the usual static assumption is relaxed, the same theory can generate both time-varying effective Newtonian gravitational constant $a_0$09 and time-varying $a_0$10, and can reproduce the redshift dependence of $a_0$11 reported in the Magneticum simulations (Tian et al., 2023).

The perturbation sector is correspondingly active. In the cosmological perturbation theory of the Blanchet–Marsat–Skordis framework, the MOND field can be interpreted as a fluid with a specific equation of state and no anisotropic stress, and baryon perturbations are shown to grow faster in the MOND regime at $a_0$12PN order. The same paper derives both $a_0$13PN and fully nonlinear exact perturbation equations and a Jeans criterion for the MOND field (Hwang et al., 2024). The khronometric analysis of the Blanchet–Marsat theory reaches a complementary conclusion: stationary MOND solutions are stable in the deep-MOND regime for several symmetries, but nonstationary low-acceleration systems generically receive order-unity khronon corrections (Flanagan, 2023).

Gravitational-wave compatibility is now a standard requirement. The Skordis–Zlosnik analysis finds that only two tensor polarizations are observable in gravitational-wave detection and that the tensor speed is exactly the speed of light (Tian et al., 2023). This sits alongside the original construction criterion that tensor modes must propagate at light speed and that the relativistic completion be free of ghost instabilities at quadratic order (Skordis et al., 2020).

Several alternative proposals state cosmological ambitions in comparable terms. Dark electromagnetism is claimed to mimic cold dark matter in CMB anisotropies, lensing, and large-scale structure because only massive particles couple to the new $a_0$14 force (Finster et al., 2023). Nonlocal metric MOND leaves the negative branch of its nonlocal function available for cosmological reconstruction (Woodard, 2014). Entropic RMOND presents galaxy-scale data fitting as a first observational check and explicitly identifies lensing, clusters, cosmology, and larger galaxy samples as the next tests (Rostami et al., 7 Nov 2025).

6. Interpretation, controversies, and open problems

A persistent misconception is that RMOND denotes a single theory. The literature instead presents a research program with sharply different ontologies: some models are metric-only, some add vectors or scalars, some use preferred foliations and explicit Lorentz-violation in the gravitational sector, some are nonlocal, and some relocate MOND from gravity to inertia or energetics. Another misconception is that a relativistic completion is needed only for galaxy rotation curves; the literature treats cosmology, lensing, and gravitational waves as equally central consistency conditions (Milgrom, 2014, Blanchet et al., 2012, Demir et al., 2014).

The status of these theories is also contested within the literature itself. Milgrom’s review argues that most full-fledged MOND theories probably are, at best, effective theories of limited applicability, and states that none had been shown to address fully the mass discrepancies in cosmology and structure formation that are otherwise explained by cosmological dark matter (Milgrom, 2014). By contrast, newer work claims agreement with the observed CMB and matter power spectra, viable background cosmologies, or CDM-like behavior in specific observables (Skordis et al., 2020, Kashfi et al., 2022, Finster et al., 2023). This suggests that cosmological viability remains the principal fault line separating different RMOND constructions.

Many proposals also acknowledge restricted domains of validity. The extended metric $a_0$15 program and the torsion-based theory derive MOND only in weak-field or MONDian regimes and explicitly do not claim to be complete theories at all curvatures (Mendoza et al., 2012, Barrientos et al., 2016). The khronometric approach achieves a consistent slow-motion limit but predicts order-unity departures from ordinary MOND for nonstationary systems and cannot guarantee stability across all accelerations (Flanagan, 2023). The brane-world picture goes further and suggests that $a_0$16 may lose meaning altogether in the relativistic regime, which, if correct, would imply that the usual MOND constant is an emergent parameter rather than a fundamental one (Milgrom, 2018).

A final unresolved issue concerns the origin of $a_0$17. Current answers include a fundamental constant in an extended metric action, a de Sitter or Hubble-scale quantity, an infrared vacuum or thermodynamic scale, a square-root-mass gauge charge, a brane balance condition, or a time-dependent local parameter. The convergence of many of these proposals on de Sitter structure, cosmic acceleration, or right-handed gauge sectors is notable, but the literature does not yet provide a unique derivation. A plausible implication is that the future of RMOND lies less in further phenomenological interpolation functions than in determining whether the MOND scale is fundamentally geometric, cosmological, quantum-vacuum, or gauge-theoretic in origin (Bernal et al., 2011, Woodard, 2014, Finster et al., 2023, Singh, 7 Jan 2026).