http://arxiv.org/abs/2003.05784
This paper introduces a possible alternative model of gravity based on fractional calculus and its applications to Newtonian gravity. In particular, Gauss’s law for gravity as well as Laplace’s equation and other fundamental classical laws are extended to a $D$-dimensional metric space, where $D$ can be a non-integer dimension. We show a possible connection between this Newtonian Fractional Gravity (NFG) and Modified Newtonian Dynamics (MOND), the leading alternative gravity model, which accounts for the observed properties of galaxies and other astrophysical structures without requiring the dark matter hypothesis. The MOND acceleration constant $a_{0} \simeq 1.2 \times 10^{ -10}\mbox{m}\thinspace \mbox{s}^{ -2}$ can be related to a natural scale length $l_{0}$ in NFG, i.e., $a_{0} \approx GM/l_{0}^{2}$, for astrophysical structures of mass $M$, and the deep-MOND regime is present in regions of space where the dimension is reduced to $D \approx 2$. For several fundamental spherically-symmetric structures, we compare MOND results such as the empirical Radial Acceleration Relation (RAR), circular speed plots, and logarithmic plots of the observed radial acceleration $g_{obs}$ vs. the baryonic radial acceleration $g_{bar}$, showing that NFG is capable of reproducing these results using a variable local dimension $D\left (w\right )$, where $w =r/l_{0}$ is a dimensionless radial coordinate. At the moment, we are unable to derive explicitly this dimension function $D\left (w\right )$ from first principles, but it can be obtained empirically in each case from the general RAR. Additional work on the subject, including studies of axially-symmetric structures, detailed galactic rotation curves fitting, and a possible relativistic extension, will be needed to establish NFG as a viable alternative model of gravity.
G. Varieschi
Fri, 13 Mar 20
48/53
Comments: 23 pages, including 4 figures