MOND-like Fractional Laplacian Theory

Abstract

I provide a derivation of some characteristic effects of Milgrom's modified Newtonian dynamics (MOND) from a fractional version of Newton's theory based on the fractional Poisson equation. I employ the properties of the fractional Laplacian to investigate the features of the fundamental solution of the proposed model. The key difference between MOND and the fractional theory introduced here is that the latter is an inherently linear theory, featuring a characteristic length scale $\ell$, whilst the former is ultimately nonlinear in nature and it is characterized by an acceleration scale $a_0$. Taking advantage of the Tully-Fisher relation, as the fractional order $s$ approaches $3/2$, I then connect the length scale $\ell$, emerging from this modification of Newton's gravity, with the critical acceleration $a_0$ of MOND. Finally, implications for galaxy rotation curves of a variable-order version of the model are discussed.

7 pages, 2 figures. Title changed. Updated to match the published version

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• Newtonian Fractional-Dimension Gravity and Disk Galaxies
• A quantum state for the late Universe
• Emergence of fractal cosmic space from fractional quantum gravity
• Gravitational potential and galaxy rotation curves in multi-fractional spacetimes
• A computational approach to exponential-type variable-order fractional differential equations
• Dark Matter in Fractional Gravity I: Astrophysical Tests on Galactic Scales
• Conformal and Non-Minimal Couplings in Fractional Cosmology
• Galaxy rotation favors prolate dark matter haloes
• Fractional Schwarzschild-Tangherlini black hole with a fractal event horizon
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• Elongated Gravity Sources as an Analytical Limit for Flat Galaxy Rotation Curves
• Newtonian Fractional-Dimension Gravity and Galaxies without Dark Matter
• Fractional entropy of the Brown-Kuchař dust in fractional anti-de Sitter quantum gravity
• Is MOND necessarily nonlinear?
• Fractional Particle and Sigma Model
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• A Relativistic Tensorial Model for Fractional Interaction between Dark Matter and Gravity
• Relativistic Fractional-Dimension Gravity
• Isothermal spheres in general relativity and Horava-type gravity
• Constraining fractionality using some observational tests