Newtonian Fractional-Dimension Gravity and the Mass-Dimension Field Equation

TL;DR

The paper advances Newtonian Fractional-Dimension Gravity (NFDG), a DM-free framework where galaxies are treated as fractal objects with a radial dimension $D(R)$. It introduces a mass-dimension field equation to derive $D_m(R)$ from baryonic mass distributions and applies both $D(R)$ and $D_m(R)$ to SPARC galaxies, including three new cases (NGC 6946, NGC 3198, NGC 2841) and revisiting three prior ones (NGC 7814, NGC 6503, NGC 3741). The results show that $D(R)$ can reproduce observed rotation curves without DM, while $D_m(R)$ provides a consistent but less accurate alternative, validating the fractal-dimension approach while highlighting sensitivity to baryonic mass modeling. These findings support NFDG as a theoretically grounded, DM-free description of galactic dynamics and point to future work expanding the galaxy sample and refining the mass-dimension framework.

Abstract

We resume our analysis of Newtonian Fractional-Dimension Gravity (NFDG), an alternative gravitational model that does not require the dark matter (DM) paradigm. We add three more galaxies (NGC 6946, NGC 3198, NGC 2841) to the catalog of those studied with NFDG methods. Once again, NFDG can successfully reproduce the observed rotation curves by using a variable fractional dimension $D\left (R\right)$, as with the nine other galaxies previously studied with these methods. In addition, we introduce a mass-dimension field equation for our model, which is capable of deriving the fractional mass dimension $D_{m}\left(R \right)$ from a single equation, as opposed to the previous $D\left (R\right)$, which was obtained simply by matching the experimental rotational velocity data for each galaxy. While the NFDG predictions computed with this new $D_{m}\left(R\right)$ dimension are not as accurate as those based on the original $D\left (R\right)$, they nevertheless confirm the validity of our fractional-dimension approach. Three previously studied galaxies (NGC 7814, NGC 6503, NGC 3741) were analyzed again with these new methods, and their structure was confirmed to be free from any dark matter components.

Figures (6)

• Figure 1: NFDG results for NGC 6946. Top panel: NFDG variable dimension $D\left (R\right )$, based directly on SPARC data (red-solid curve), compared with NFDG mass-dimension $D_{m}\left(R\right)$ (blue-dashed curve), and fixed values $D =2.3-2.5$ (black-dotted lines). Bottom panel: NFDG rotation curves (circular velocity vs. radial distance) compared to the original SPARC data (black circles with error bars). The NFDG best fit for the variable dimension $D\left (R\right )$ is shown by the red-solid line, while the NFDG fit for the mass-dimension $D_{m}\left(R\right)$ is shown by the blue-dashed curve. Also shown: MOND prediction based on the general RAR (green, dot-dashed), Newtonian rotation curves (different components - gray lines, total - black dashed line), and asymptotic flat velocity band (horizontal gray band).
• Figure 2: NFDG results for NGC 3198. Top panel: NFDG variable dimension $D\left (R\right )$, based directly on SPARC data (red-solid curve), compared with NFDG mass-dimension $D_{m}\left(R\right)$ (blue-dashed curve), and fixed values $D =1.8-2.0$ (black-dotted lines). Bottom panel: NFDG rotation curves (circular velocity vs. radial distance) compared to the original SPARC data (black circles with error bars). The NFDG best fit for the variable dimension $D\left (R\right )$ is shown by the red-solid line, while the NFDG fit for the mass-dimension $D_{m}\left(R\right)$ is shown by the blue-dashed curve. Also shown: MOND prediction based on the general RAR (green, dot-dashed), Newtonian rotation curves (different components - gray lines, total - black dashed line), and asymptotic flat velocity band (horizontal gray band).
• Figure 3: NFDG results for NGC 2841. Top panel: NFDG variable dimension $D\left (R\right )$, based directly on SPARC data (red-solid curve), compared with NFDG mass-dimension $D_{m}\left(R\right)$ (blue-dashed curve), and fixed values $D =1.9-2.1$ (black-dotted lines). Bottom panel: NFDG rotation curves (circular velocity vs. radial distance) compared to the original SPARC data (black circles with error bars). The NFDG best fit for the variable dimension $D\left (R\right )$ is shown by the red-solid line, while the NFDG fit for the mass-dimension $D_{m}\left(R\right)$ is shown by the blue-dashed curve. Also shown: MOND prediction based on the general RAR (green, dot-dashed), Newtonian rotation curves (different components - gray lines, total - black dashed line), and asymptotic flat velocity band (horizontal gray band).
• Figure 4: NFDG results for NGC 7814. Top panel: NFDG variable dimension $D\left (R\right )$, based directly on SPARC data (red-solid curve), compared with NFDG mass-dimension $D_{m}\left(R\right)$ (blue-dashed curve), and fixed values $D =2.3-2.5$ (black-dotted lines). Bottom panel: NFDG rotation curves (circular velocity vs. radial distance) compared to the original SPARC data (black circles with error bars). The NFDG best fit for the variable dimension $D\left (R\right )$ is shown by the red-solid line, while the NFDG fit for the mass-dimension $D_{m}\left(R\right)$ is shown by the blue-dashed curve. Also shown: MOND prediction based on the general RAR (green, dot-dashed), Newtonian rotation curves (different components - gray lines, total - black dashed line), and asymptotic flat velocity band (horizontal gray band).
• Figure 5: NFDG results for NGC 6503. Top panel: NFDG variable dimension $D\left (R\right )$, based directly on SPARC data (red-solid curve), compared with NFDG mass-dimension $D_{m}\left(R\right)$ (blue-dashed curve), and fixed values $D =1.9-2.1$ (black-dotted lines). Bottom panel: NFDG rotation curves (circular velocity vs. radial distance) compared to the original SPARC data (black circles with error bars). The NFDG best fit for the variable dimension $D\left (R\right )$ is shown by the red-solid line, while the NFDG fit for the mass-dimension $D_{m}\left(R\right)$ is shown by the blue-dashed curve. Also shown: MOND prediction based on the general RAR (green, dot-dashed), Newtonian rotation curves (different components - gray lines, total - black dashed line), and asymptotic flat velocity band (horizontal gray band).