Newtonian Fractional-Dimension Gravity

• NFDG is a gravitational framework that extends Newtonian laws by incorporating a variable, non-integer effective spatial dimension to describe galactic dynamics.
• The method computes gravitational fields using fractional-dimensional metric spaces, with D(R) transitioning from nearly 3 in dense regions to close to 2 in low-acceleration outer areas.
• Applications of NFDG include accurately reproducing rotation curves and empirical dimension profiles in galaxies such as NGC 6503, the Milky Way, and others without invoking dark matter.

Newtonian Fractional-Dimension Gravity (NFDG) is a Newtonian-like alternative gravity framework in which the effective spatial dimension entering gravitational dynamics is allowed to be non-integer, with $1 \le D \le 3$, and in astrophysical applications may vary with position, typically as a radial function $D(R)$. Its central use is galactic dynamics without dark matter: the baryonic mass distribution is retained, while the gravitational field is computed as if the relevant geometry were a fractional-dimensional metric space rather than fixed 3-dimensional Euclidean space. In later work the label “Fractional-Dimension Gravity” (FDG) is adopted for the same program, including tentative relativistic extensions (Varieschi, 2020, Varieschi, 2020, Varieschi, 1 Jul 2026).

1. Conceptual basis

NFDG was introduced as an extension of the laws of Newtonian gravitation to lower dimensional spaces, including those with fractional dimension, and it is explicitly presented as distinct from fractional-derivative gravity. The “fractional” ingredient is the effective Hausdorff dimension of the matter distribution, not the replacement of Newtonian field equations by nonlocal operators of fractional order. In this sense, the theory is formulated on a non-integer-dimensional geometry while retaining integer-order, local differential structure (Varieschi, 2020, Varieschi, 2021).

The framework is motivated by the possibility that galactic structures behave as fractal media. In the working interpretation used across the NFDG literature, the effective dimension is close to $D \simeq 3$ in Newtonian inner regions and approaches $D \simeq 2$ in low-acceleration outer regions, where flat rotation curves occur. The theory is therefore not dark matter, not exactly MOND, and not a “fractional MOND” theory; it is a MOND-like phenomenological framework in which dark matter phenomenology is recast as dimensional phenomenology (Varieschi, 2020, Varieschi, 2022).

A recurrent clarification is that NFDG does not imply a literal change in the tridimensionality of physical space inside galaxies. Rather, the local Hausdorff-type dimension $D\neq 3$ is associated with the matter distribution or with the effective geometry relevant to the gravitational field. This distinction is central to the way NFDG is separated from both standard dark-halo modeling and operator-based fractional gravities (Pascoli, 2024, Varieschi, 2020).

2. Mathematical framework

The formalism introduces a natural length scale $l_0$ and dimensionless coordinates $\mathbf{w}=\mathbf{x}/l_0$. In the constant-$D$ theory, the generalized Gauss law gives the point-mass field magnitude

$$|\mathbf g| = 2\pi^{1-D/2}\Gamma(D/2)\frac{G m_{(D)}}{r^{D-1}},$$

so the inverse-square law becomes an inverse-$(D-1)$-power law. The limiting cases emphasized in the literature are $D(R)$0, which reproduces standard Newtonian gravity, and $D(R)$1, which yields an inverse-linear field and thereby the deep-MOND-like regime (Varieschi, 2020).

In the notation used in the galaxy papers, the point-mass potential is

$D(R)$2

while for $D(R)$3,

$D(R)$4

The corresponding radial field and circular speed are written as

$D(R)$5

These formulas reduce to the standard Newtonian relations when $D(R)$6 (Varieschi, 2022).

For extended sources, the potential in dimensionless coordinates is

$D(R)$7

with the logarithmic Green kernel at $D(R)$8. The generalized Poisson equation is

$D(R)$9

and the $D \simeq 3$0-dimensional measure for a spherically symmetric function is

$D \simeq 3$1

These relations supply the measure-theoretic basis for NFDG as gravity in a non-integer-dimensional space (Varieschi, 2020, Varieschi, 2021).

3. Extended sources, disks, and reconstruction of $D \simeq 3$2

The extension from spherical systems to realistic galaxies is built on the generalized Green kernel

$D \simeq 3$3

with Gegenbauer polynomials $D \simeq 3$4. This multipole expansion is the computational core for spherical bulges and cylindrical disk or gas components. Earlier work used six nonzero terms $D \simeq 3$5, while later work reports that using only the first four nonzero terms $D \simeq 3$6 produces differences of typically less than $D \simeq 3$7 (Varieschi, 2020, Varieschi, 26 Jul 2025).

For disk galaxies, NFDG is applied to the baryonic decomposition into bulge, stellar disk, and gas. In practical calculations the effective dimension is taken to be a function of cylindrical radius,

$D \simeq 3$8

and spatial derivatives of $D \simeq 3$9 are neglected when taking the gradient of the potential. This approximation is repeatedly identified as a central working assumption of the method (Varieschi, 2022, Varieschi, 26 Jul 2025).

The empirical reconstruction of $D \simeq 2$0 is straightforward. One computes the observed acceleration $D \simeq 2$1 from rotation-curve data, computes the NFDG field $D \simeq 2$2 from the known baryonic mass model, and solves at each radius

$D \simeq 2$3

In one later implementation, the interval $D \simeq 2$4 is sampled at 50 equally spaced points,

$D \simeq 2$5

and the resulting $D \simeq 2$6 values are interpolated to obtain a continuous dimension profile. Because $D \simeq 2$7 is reconstructed point-by-point from the observed acceleration, the resulting NFDG rotation curve is expected to be essentially perfect (Varieschi, 26 Jul 2025).

A more predictive but less accurate alternative is the mass-dimension proxy

$D \simeq 2$8

with

$D \simeq 2$9

Later work replaces this proxy by a mass-dimension field equation intended to derive $D\neq 3$0 from first principles, but the resulting rotation curves remain less accurate than those obtained from the empirical $D\neq 3$1 reconstruction (Varieschi, 2022, Varieschi, 26 Jul 2025).

4. Astrophysical applications

NFDG was first developed for spherical systems, then for axially symmetric thin and thick disks, and then applied to rotationally supported galaxies such as NGC 6503, NGC 7814, and NGC 3741. In these studies, the empirical Radial Acceleration Relation and individual SPARC rotation curves are both used to infer a galaxy-specific $D\neq 3$2, with the outer parts often approaching the MOND-like regime $D\neq 3$3 (Varieschi, 2020, Varieschi, 2020).

Subsequent work expanded the sample to NGC 5033, NGC 6674, NGC 5055, and NGC 1090, again without any dark matter component. The dimension profiles reported there follow a common pattern: inner values near $D\neq 3$4, and outer values typically in the range $D\neq 3$5–$D\neq 3$6, although the gas-dominated dwarf NGC 3741 is reported to remain around $D\neq 3$7–$D\neq 3$8 over much of its radial range. The same paper estimates an uncertainty in the inferred $D\neq 3$9 of roughly $l_0$0 over most of the radial range from the extended NGC 5055 analysis (Varieschi, 2022).

A broader set of case studies extends NFDG beyond ordinary spiral disks. For AGC 114905, a variable dimension in the range $l_0$1 reproduces the observed rotation curve, but a fixed $l_0$2 curve still fits all experimental data within their error bars. For NGC 1052-DF2, a virial-theorem argument yields $l_0$3, consistent with nearly Newtonian behavior. For the Bullet Cluster merger, a simplified NFDG treatment explains the observed infall velocity with $l_0$4 and without dark matter (Varieschi, 2022).

More recent work adds NGC 6946, NGC 3198, and NGC 2841, and reports that NFDG can successfully reproduce the observed rotation curves by using a variable $l_0$5. The same study reanalyzes NGC 7814, NGC 6503, and NGC 3741 with the new mass-dimension methods and states that their structure is free from any dark matter components (Varieschi, 26 Jul 2025). Applied to the Milky Way with Gaia DR3-based and earlier rotation-curve data, the renamed FDG framework finds outer-disk values roughly $l_0$6, with $l_0$7 near the Solar radius (Varieschi, 1 Jul 2026).

System	Reported NFDG/FDG dimension	Application
AGC 114905	$l_0$8	Rotation curve
NGC 1052-DF2	$l_0$9	Velocity-dispersion/virial estimate
Bullet Cluster	$\mathbf{w}=\mathbf{x}/l_0$0	Infall velocity
Milky Way	$\mathbf{w}=\mathbf{x}/l_0$1	Rotation curves

These case studies collectively suggest that NFDG is being used as a galaxy-by-galaxy or system-by-system dimensional phenomenology rather than as a universal one-parameter law (Varieschi, 2022, Varieschi, 1 Jul 2026).

5. Relation to MOND and distinction from other fractional gravities

The MOND connection is explicit. NFDG relates the MOND acceleration scale to the geometric scale length by

$\mathbf{w}=\mathbf{x}/l_0$2

and interprets the deep-MOND regime as the limit $\mathbf{w}=\mathbf{x}/l_0$3. In this regime the field scales as $\mathbf{w}=\mathbf{x}/l_0$4, the potential is logarithmic, and the asymptotic circular speed becomes nearly constant, reproducing the baryonic Tully–Fisher scaling $\mathbf{w}=\mathbf{x}/l_0$5 (Varieschi, 2020, Varieschi, 2020).

At the same time, NFDG differs structurally from MOND. It does not introduce a MOND interpolation law as a fundamental postulate; the controlling quantity is the local dimension profile $\mathbf{w}=\mathbf{x}/l_0$6, or in later work its baryon-driven counterpart $\mathbf{w}=\mathbf{x}/l_0$7, rather than an acceleration threshold. A further distinction concerns the External Field Effect. One NFDG study argues that for $\mathbf{w}=\mathbf{x}/l_0$8 the shell theorem still holds in the fractional-dimensional measure, so NFDG does not imply an External Field Effect in that interval; the no-EFE claim is explicitly restricted to $\mathbf{w}=\mathbf{x}/l_0$9, and the author notes that $D$0 could behave differently (Varieschi, 2022).

NFDG is also distinct from several other “fractional gravity” programs. Operator-based fractional Newtonian gravity replaces the ordinary Laplacian with a fractional Laplacian $D$1, so the fractional ingredient is a nonlocal operator rather than a non-integer spatial dimension (Giusti et al., 2020, Ourabah, 2024). Other approaches use Riemann–Liouville fractional derivatives in relativistic field equations, or Riesz-type power-law kernels in ordinary $D$2, again without reformulating Newtonian gravity as a theory on a fractional-dimensional space (Munkhammar, 2010, Varieschi, 2017). This distinction is repeatedly emphasized in the NFDG and RFDG literature (Varieschi, 2021).

6. Relativistic extensions, nomenclature, and open issues

A relativistic extension, “Relativistic Fractional-Dimension Gravity” (RFDG), carries the same measure-theoretic idea into a weighted Hilbert action

$D$3

with the simplest FDG-motivated choice $D$4, $D$5. The resulting field equations contain the extra geometric terms

$D$6

and a cosmological application uses a purely temporal weight

$D$7

to derive modified Friedmann equations. This relativistic program is presented as exploratory rather than definitive (Varieschi, 2021).

The limitations of the Newtonian theory are repeatedly acknowledged. Early papers state that $D$8 is not derived from first principles. The direct inversion method provides excellent fits but only by treating $D$9 as an empirical reconstruction. The mass-dimension field equation is intended to address this by deriving $$|\mathbf g| = 2\pi^{1-D/2}\Gamma(D/2)\frac{G m_{(D)}}{r^{D-1}},$$0 from the baryonic distribution, yet later work explicitly reports that $$|\mathbf g| = 2\pi^{1-D/2}\Gamma(D/2)\frac{G m_{(D)}}{r^{D-1}},$$1-based predictions are not as accurate as those based on the original $$|\mathbf g| = 2\pi^{1-D/2}\Gamma(D/2)\frac{G m_{(D)}}{r^{D-1}},$$2 (Varieschi, 2020, Varieschi, 26 Jul 2025).

Recent work adopts the shorter label FDG and applies it to the Milky Way, while also proposing speculative implications of the fractional metric for special relativity. One such proposal is the possibility of effective superluminal motion in galactic regions where $$|\mathbf g| = 2\pi^{1-D/2}\Gamma(D/2)\frac{G m_{(D)}}{r^{D-1}},$$3. The paper itself describes this result as “very speculative,” and it is best regarded as a conjectural extension of the framework rather than an established consequence (Varieschi, 1 Jul 2026).

Taken together, the literature presents NFDG as a mathematically structured but still incomplete alternative to dark matter on galactic scales: concrete enough to generate detailed rotation-curve constructions and dimension profiles, but not yet equipped with a universally predictive law for $$|\mathbf g| = 2\pi^{1-D/2}\Gamma(D/2)\frac{G m_{(D)}}{r^{D-1}},$$4, a settled relativistic completion, or a fully developed first-principles derivation of the effective fractional dimension.