Newtonian Fractional-Dimension Gravity and MOND

Abstract

We continue our analysis of Newtonian Fractional-Dimension Gravity (NFDG), an extension of the standard laws of Newtonian gravity to lower dimensional spaces including those with fractional (i.e., non-integer) dimension. We apply our model to three rotationally supported galaxies: NGC 7814 (Bulge-Dominated Spiral), NGC 6503 (Disk-Dominated Spiral), and NGC 3741 (Gas-Dominated Dwarf). As was done in the general cases of spherically-symmetric and axially-symmetric structures, which were studied in previous work on the subject, we examine a possible connection between NFDG and Modified Newtonian Dynamics (MOND), a leading alternative gravity model which explains the observed properties of these galaxies without requiring the Dark Matter (DM) hypothesis. In NFDG, the MOND acceleration constant a0 1.2 × 10 −10 m s −2 can be related to a natural scale length l0, namely a0 ≈ GM/l 2 0 for a galaxy of mass M. Also, the empirical Radial Acceleration Relation (RAR), connecting the observed radial acceleration g obs with the baryonic one g bar , can be explained in terms of a variable local dimension D. As an example of this methodology, we provide detailed rotation curve fits for the three galaxies mentioned above.

Relativistic Fractional-Dimension Gravity

“…NFDG also shows possible connections with Modified Newtonian Dynamics (MOND) [21][22][23], as discussed in detail in our previous papers [1][2][3]. In particular, MOND phenomenology, including the recently reported Radial Acceleration Relation (RAR) [24][25][26], might be explained by our varying dimension D = D(r), which provides the link between the inherently non-linear MOND theory and the linear NFDG.…”

“…where the infinitesimal measures dµ i in the second line of the previous equations follow from the original Stillinger's weight W 1 described above, and used in the integral in Equation (1). The factor of two in the weight W 1 is now omitted, assuming integration between −∞ and +∞ in each sub-space X i .…”

“…This paper considers a possible relativistic generalization of Newtonian Fractional-Dimension Gravity (NFDG), which was previously introduced as a non-relativistic alternative gravity model ( [1][2][3], papers I, II, and III in the following). The main goal of NFDG was to model galactic rotation curves without using the controversial Dark Matter (DM) component (see also [4] for a general overview of NFDG).…”

“…This paper considers a possible relativistic generalization of Newtonian Fractional-Dimension Gravity (NFDG), which was previously introduced as a non-relativistic alternative gravity model ( [1][2][3], papers I, II, and III in the following). The main goal of NFDG was to model galactic rotation curves without using the controversial Dark Matter (DM) component (see also [4] for a general overview of NFDG). This was done by assuming that galactic structures might behave like fractal media, with an effective spatial dimension which could be lower than the standard value D = 3, including possible fractional, i.e., non-integer values.…”

“…In paper I [1], it was shown how NFDG is a natural extension of standard Newtonian gravity to non-integer dimension spaces. Starting from a heuristic extension of Gauss's law for gravity to fractional-dimension spaces, we were able to generalize the gravitational field and potential for extended mass sources, the Laplace and Poisson equations, the multipole expansion, etc.…”