FRACTIONAL-DIMENSION GRAVITY (FDG)

INTRODUCTION TO FDG

Fractional Dimension Gravity (FDG) is an alternative model of gravity that I introduced in 2020-21, with the first three papers on the subject (see links at the top of this page). The original name was Newtonian Fractional-Dimension Gravity (NFDG), but in my 8th paper on the subject, I decided to simplify the name to FDG. The main goal of FDG is to model galactic rotation curves without using the controversial Dark Matter (DM). This is done by assuming that galactic structures behave as fractal media, with an effective dimension which can be lower than the standard value D=3. This includes also possible fractional, i.e. non-integer, dimensions. This model is broadly based on the methods of fractional mechanics and fractional calculus. It is also connected with Modified Newtonian Dynamics (MOND), a leading alternative theory of gravity introduced by M. Milgrom in 1983.

In the first paper (published version), I used Newtonian FDG to model several types of spherically-symmetric structures, while in the second paper (published version) I did the same for axially-symmetric structures. However, the real objective of FDG is to model existing galactic rotation curves, obtained from the SPARC database or others. In the second paper, I produced the first Newtonian FDG example of detailed data fitting, for the disk-dominated galaxy NGC 6503 (see photo, NASA/ESA Hubble Space Telescope).

In the figure below, I show the latest Newtonian FDG analysis of this galaxy (from the 7th paper - 2025). The top panel shows the variable dimension D as a function of the radial distance R (red-solid curve), computed with the main NFDG model. This dimension is very close to a constant value D=2.0-2.1 over most of the radial range. This is typical NFDG behavior for galactic structures dominated by an axially-symmetric stellar disk component. Bottom panel: NFDG galactic rotation curves compared with SPARC data. There is perfect agreement between the main NFDG curve (red-solid) and the experimental data. This shows that, if this galaxy behaves as a fractal structure whose dimension D follows the red-solid curve in the top panel, the resulting NFDG circular velocity curve in the bottom panel (red-solid) will perfectly fit the experimental data, without any need of dark matter. See also my third, fifth, and seventh papers for a full discussion of this galaxy.

In the third paper (published version) on the subject, I extended the Newtonian FDG analysis to two additional galaxies: NGC 7814 and NGC 3741. A fourth paper introduced a relativistic extension of the model: Relativistic Fractional-Dimension Gravity (RFDG), which might be useful for astrophysical or cosmological applications. A fifth paper analyzed four more galaxies (NGC 5033, NGC 6674, NGC 5055, NGC 1090), in relation with the so-called External Field Effect (EFE): NFDG and the External Field Effect. In a sixth paper, NFDG and Galaxies without Dark Matter, Newtonian FDG was applied to some notable cases of galaxies with little or no dark matter, such as AGC 114905 and NGC 1052-DF2.

In the seventh FDG paper, three more galaxies (NGC 6946, NGC 3198, NGC 2841) were added to the catalog of those studied with FDG methods. Once again, Newtonian FDG was successful in reproducing the observed rotation curves by using the variable fractional dimension D(R). In addition, a new FDG field equation was introduced, which is capable of deriving the so-called mass-dimension D_m(R) from first principles, as opposed to the previous D(R) which was obtained simply by matching the experimental rotational velocity data for each galaxy. While the FDG predictions computed with this new D_m(R) dimension are not as accurate as those based on the original D(R), they nevertheless confirm the validity of our fractional-dimension approach. An example of this new method, based on the mass-dimension D_m(R), can be seen in the figure above for NGC 6503 (blue-dashed curves in both panels).

Finally, in the most recent eight FDG paper, our home galaxy, the Milky Way, was also studied successfully with FDG methods and some potential consequences for motions in fractional-dimension spaces were analyzed.

All current FDG galactic fits obtained with these new methods are detailed in the links at the bottom of this page, for several different galaxies of the SPARC database. New fits will be added as they become available. In all these cases, the galactic structures were confirmed to be free from any dark matter components.