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.

中文翻译：

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牛顿分数维重力和 MOND

本文介绍了一种基于分数维空间理论的可能的替代引力模型及其在牛顿引力中的应用。特别是，高斯万有引力定律以及其他基本经典定律被扩展到一个 $D$ 维的度量空间，其中 $D$ 可以是一个非整数维度。我们展示了这种牛顿分数维引力 (NFDG) 和修正牛顿动力学 (MOND) 之间可能存在的联系，这是一种领先的替代引力模型，它解释了星系和其他天体物理结构的观察特性，而无需暗物质假设。MOND 加速度常数 $a_{0} \simeq 1.2 \times 10^{ -10}\mbox{m}\thinspace \mbox{s}^{ -2}$ 可以与自然尺度长度 $l_{0 }$ 在 NFDG，即 $a_{0} \approx GM/l_{0}^{2}$，对于质量为 $M$ 的天体物理结构，并且深 MOND 机制存在于维度减少到 $D \大约 2$ 的空间区域中。对于几个基本的球对称结构，我们比较了 MOND 结果，例如经验径向加速度关系 (RAR)、圆周速度图以及观测到的径向加速度 $g_{obs}$ 与重子径向加速度 $g_ 的对数图{bar}$，带有 NFDG 结果。我们表明我们的模型能够使用可变局部维度 $D\left (w\right )$ 重现这些结果，其中 $w =r/l_{0}$ 是无量纲的径向坐标。目前，我们无法从第一性原理中明确推导出这个维度函数 $D\left (w\right )$，但可以在每种情况下从一般 RAR 中凭经验获得。关于这个主题的额外工作，