[Paper Review] Modified Newtonian Dynamics In Dimensionless Form

W. F. Kao|arXiv (Cornell University)|Apr 1, 2005

Computational Physics and Python Applications3 citations

TL;DR

This paper reformulates Modified Newtonian Dynamics (MOND) in dimensionless form using scale-invariant units derived from a critical acceleration $g_0$, galaxy mass $M_0$, and radius $r_0$, enabling a unified description of gravitational fields across short- and long-distance regimes. It introduces a scale-dependent effective dimension $d(r)$ that transitions from 3D at small scales to 2D at large scales, with a smooth $oldsymbol{\Lambda}$-shaped function resolving singularities at $r=1$, offering a geometric interpretation of MOND's phenomenology.

ABSTRACT

Modified Newtonian dynamics proposed that gravitational field needs modifications when the field strength $g$ is weaker than a critical value $g_0$. This has been shown to be a good candidate as an alternative to cosmic dark matter. There is another way to look at this theory as a length scale dependent theory. One will show that modification of the Newtonian field strength depends on the mass distribution and the coordinate scale of the system. It is useful to separate the effective gravitation field $g(r)$ into a small scale (or short-distance)$g_s$ field and a large scale (or a long-distance) $g_l$ field that should be helpful for a better understanding of the underlying physics. The effective potential is also derived.

Motivation & Objective

• To reformulate Modified Newtonian Dynamics (MOND) in dimensionless form using physical scale parameters $g_0$, $M_0$, and $r_0$ to unify gravitational behavior across distance scales.
• To decompose the effective gravitational field $g(r)$ into short-distance ($g_s$) and long-distance ($g_l$) components for improved physical insight.
• To define and analyze an effective dimension $d(r)$ that transitions from 3D to 2D as scale increases, reflecting MOND’s observed $1/r$ and $1/r^2$ behaviors.
• To explore connections between MOND’s scale dependence and higher-dimensional theories such as Kaluza-Klein, suggesting possible geometric underpinnings.

Proposed method

• Expresses MOND’s field equation $g \cdot \mu(g/g_0) = g_N$ in dimensionless form by scaling $g$, $r$, and $m$ with $g_0$, $r_0$, and $M_0$, respectively.
• Derives the dimensionless field equation ${g^2}/{\sqrt{1+g^2}} = m/r^2$, where $g$, $m$, and $r$ are now unitless quantities.
• Introduces a characteristic length $r_c = r_0 \sqrt{m}$ to reparameterize the radial coordinate into a dimensionless variable.
• Defines the effective dimension $d(r)$ via $g \propto r^{-(d-1)}$, leading to $d(r) = 3 - \frac{\ln\left(\frac{\sqrt{1+\sqrt{1+4r^4}}}{\sqrt{2}}\right) - \ln\lambda(r)}{\ln r}$, with $\lambda(r)$ ensuring smoothness at $r=1.$
• Constructs a $\Lambda$-shaped function $\lambda(r) = 1 + \left(\frac{\sqrt{1+\sqrt{5}}}{\sqrt{2}} - 1\right)\sqrt{r}e^{1-\sqrt{r}}$ that peaks at $r=1$ and approaches 1 at $r=0$ and $r\to\infty$, resolving the singularity in $d(r)$.
• Plots $d(r)$ and compares $g(r)$ with $\lambda(r)$ to visualize the scale-dependent geometric behavior of gravity in MOND.

Experimental results

Research questions

• RQ1 How can MOND’s gravitational field be expressed in a fully dimensionless form using fundamental scale parameters?
• RQ2 What is the physical significance of separating the gravitational field into short- and long-distance components $g_s$ and $g_l$?
• RQ3 How does the effective dimension $d(r)$ of spacetime vary with scale in MOND, and can it be made mathematically well-defined across all $r$?
• RQ4 Can the observed $1/r$-like gravity at large scales in spiral galaxies be interpreted as a geometric transition to a 2D-like effective dimension?
• RQ5 What role might Kaluza-Klein-type compactification play in explaining MOND’s scale-dependent behavior?

Key findings

• The dimensionless MOND field equation ${g^2}/{\sqrt{1+g^2}} = m/r^2$ is derived, where $g$, $m$, and $r$ are scaled by $g_0$, $M_0$, and $r_0$, respectively, enabling universal application across systems.
• The effective gravitational field is decomposed into short-distance ($g_s$) and long-distance ($g_l$) components, providing a clearer physical interpretation of MOND’s scale dependence.
• An effective dimension $d(r)$ is defined such that $g \propto r^{-(d-1)}$, with $d \to 3$ at small $r$ and $d \to 2$ at large $r$, reflecting the transition from 3D to 2D gravity.
• A $\Lambda$-shaped function $\lambda(r)$ is introduced to resolve the singularity in $d(r)$ at $r=1$, ensuring the function is well-defined everywhere.
• The resulting $d(r)$ function, plotted in Fig. 8, shows a smooth transition from 3D to 2D behavior, suggesting a geometric origin for MOND’s phenomenology.
• The effective potential is derived and shown to be related to the stiffness of the distorted geometry, with $g(r)$ and $\lambda(r)$ plotted together in Fig. 7 for comparison.