[Paper Review] The Dark Mass Problem

Ll. Bel|arXiv (Cornell University)|Mar 21, 2013

Relativity and Gravitational Theory3 references3 citations

TL;DR

This paper proposes a modified Newtonian gravity law that adds a $1/r$-dependent attractive force to Newton's inverse-square law, introducing a new constant $G'$. The modification generates an effective, non-local dark matter-like density distribution, explaining galactic rotation curves and predicting novel effects such as gravitational attraction in empty cavities—offering a potential solution to the dark mass problem without requiring unseen matter.

ABSTRACT

I discuss some of the basic properties of a potential theory derived from a modified Newton's law of action at a distance that includes a $1/r$ attractive force.

Motivation & Objective

• To address the dark mass problem by proposing a modified gravitational force law that includes an additional $1/r$-dependent attractive term.
• To demonstrate that this modified force law naturally produces an effective, positive density distribution resembling dark matter, even in regions with no ordinary matter.
• To explore the cosmological and astrophysical implications of this non-local gravitational theory, particularly in explaining galactic rotation curves and lensing effects.
• To show that the theory can be consistently extended to General Relativity by replacing the matter density with an effective density in the Einstein equations.
• To investigate the physical consequences of the theory, including the 'empty cavity effect'—where massive bodies are attracted toward centers with no actual mass.

Proposed method

• Derive a modified force law between point masses as $ F^i = -G m_1 m_2 (x^i - y^i)/r^3 - G' m_1 m_2 (x^i - y^i)/r^2 $, introducing a new constant $ G' $ with dimensions of $ M^{-1}L^2T^{-2} $.
• Define a gravitational potential $ V(x) $ such that $ F^i = -m_1 rac{ abla V}{ abla x^i} $, leading to $ V(x) = -G rac{m_2}{r} + G' m_2 rac{ ext{ln}(r)}{r} $ for point sources.
• Use Poisson's equation $ abla^2 V = 4 ilde{ ho} $ to derive an effective density $ ho(x) = ilde{ ho}(x) $, where $ ilde{ ho}(x) = ho_{ ext{Newton}} + ext{non-local term} $, with $ ho_{ ext{Newton}} $ being the standard mass density.
• For extended sources with constant density $ ho $ in a sphere or spherical shell, compute the effective density $ ho_{ ext{eff}}(r) $ using integrals involving $ rac{1}{r^2} $ and the function $ ho_{ ext{eff}}(r) = ho H(a-r) + 2 ilde{ ho} ilde{ ho} ext{ with } ilde{ ho} = rac{1}{4 ilde{ ho}} rac{G'}{G} $.
• Analyze the case of a spherical cavity (inner radius $ b $, outer $ a $) to show that the effective density remains positive even when $ ho = 0 $, leading to gravitational attraction toward the center.
• Extend the theory to General Relativity by replacing the standard energy-momentum tensor source with the effective density $ ho(x) $, showing that vacuum solutions become approximate solutions and point-particle singularities become more singular.

Experimental results

Research questions

• RQ1 Can a modified Newtonian gravity law with an additional $1/r$-dependent force explain the observed flat rotation curves of spiral galaxies without invoking dark matter?
• RQ2 Does the inclusion of a non-local, $1/r^2$-weighted contribution to the gravitational potential lead to a positive effective mass density in regions with no actual matter?
• RQ3 What are the implications of such a modified gravity theory for the structure of voids and the behavior of massive bodies in empty cavities?
• RQ4 How can this modified gravity theory be consistently embedded into the framework of General Relativity?
• RQ5 Does this theory predict the 'empty cavity effect', where gravitational attraction persists in spatial regions devoid of matter?

Key findings

• The modified force law introduces a non-local effective density $ ho_{ ext{eff}}(r) = ho H(a-r) + 2 ilde{ ho} ilde{ ho} ext{ with } ilde{ ho} = rac{1}{4 ilde{ ho}} rac{G'}{G} $, which remains positive even in regions with zero ordinary matter density.
• In a spherical cavity (with $ r < b $), the effective density is non-zero and positive, leading to a net attractive force toward the center despite the absence of physical mass—demonstrating the 'empty cavity effect'.
• The theory predicts that the gravitational field of a point particle becomes more singular than in standard Einstein gravity, due to the non-local nature of the effective density.
• All exact vacuum solutions of General Relativity become approximate solutions in this modified theory, since the effective density $ ho_{ ext{eff}} $ is non-zero even in vacuum regions.
• The theory naturally explains galactic rotation curves and weak lensing effects by generating a dark matter-like gravitational potential without requiring new particles.
• The proposed potential $ V(x) = -G rac{m_2}{r} + G' m_2 ext{ln}(r) $ leads to a force law that, while not equivalent to other $ ext{ln}(r) $-based models, shares the key feature of making point sources appear spatially distributed, consistent with the idea of 'loss of point-like character'.