Newtonian Fractional-Dimension Gravity and MOND

Abstract

This paper introduces a possible alternative model of gravity based on the theory of fractional-dimension spaces and its applications to Newtonian gravity. In particular, Gauss's law for gravity as well as 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-Dimension Gravity (NFDG) and Modified Newtonian Dynamics (MOND), a 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 NFDG, 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 $g_{bar}$, with NFDG results. We show that our model 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 NFDG as a viable alternative model of gravity.

25 pages, including 4 figures: change of title and other minor changes after peer review. Final version to be published in Foundations of Physics, Springer

References in corpus (12)

• SPARC: Mass Models for 175 Disk Galaxies with Spitzer Photometry and Accurate Rotation Curves
• The Radial Acceleration Relation in Rotationally Supported Galaxies
• One Law To Rule Them All: The Radial Acceleration Relation of Galaxies
• A new relativistic theory for Modified Newtonian Dynamics
• The modified Newtonian dynamics-MOND-and its implications for new physics
• Vector Calculus in Non-Integer Dimensional Space and its Applications to Fractal Media
• Anisotropic Fractal Media by Vector Calculus in Non-Integer Dimensional Space
• Multifractional theories: an unconventional review
• MOND-like Fractional Laplacian Theory
• The Third Law of Galactic Rotation
• What is the price of abandoning dark matter? Cosmological constraints on alternative gravity theories
• Newtonian Fractional-Dimension Gravity and Disk Galaxies

Cited by in corpus (12)

• Newtonian Fractional-Dimension Gravity and Rotationally Supported Galaxies
• Newtonian Fractional-Dimension Gravity and Disk Galaxies
• Emergence of fractal cosmic space from fractional quantum gravity
• Gravitational potential and galaxy rotation curves in multi-fractional spacetimes
• Dark Matter in Fractional Gravity I: Astrophysical Tests on Galactic Scales
• Elongated Gravity Sources as an Analytical Limit for Flat Galaxy Rotation Curves
• Newtonian Fractional-Dimension Gravity and the External Field Effect
• Newtonian Fractional-Dimension Gravity and Galaxies without Dark Matter
• Fractional entropy of the Brown-Kuchař dust in fractional anti-de Sitter quantum gravity
• The $\kappa$-model under test of the SPARC database
• Isothermal spheres in general relativity and Horava-type gravity
• Relativistic Fractional-Dimension Gravity