[Paper Review] On the problem of anisotropy in geometrodynamics

Sergey Siparov|arXiv (Cornell University)|Sep 10, 2008

Advanced Differential Geometry Research1 references3 citations

TL;DR

This paper proposes a modification of the Einstein-Hilbert action in geometrodynamics that introduces anisotropic terms, leading to generalized geodesic equations and a gravity force law with non-Newtonian contributions. The approach preserves the equivalence principle, explains flat rotation curves of spiral galaxies, and reproduces the Tully-Fisher relation and the 3D gravitational well without dark matter.

ABSTRACT

The problem of spiral galaxies rotation curves and some others are approached classically on the base of the geometrodynamics. The modification of the expression for Einstein-Hilbert action leads to the generalized geodesics and then to the equations for the gravity force that contain not only the Newtonian term. The used approach is consistent with the equivalence principle, preserves all the results of classical geometrodynamics, and provides an explanation for flat rotation curves of the spiral galaxies together with the 3D gravitational well problem and Tully-Fisher law.

Motivation & Objective

• To resolve the discrepancy between observed flat rotation curves in spiral galaxies and predictions from classical general relativity.
• To address the 3D gravitational well problem, where standard models fail to account for the depth of gravitational potential in galaxies.
• To provide a geometric explanation for the Tully-Fisher law, linking luminosity to rotational velocity, within a modified geometrodynamical framework.
• To maintain consistency with the equivalence principle while introducing anisotropic corrections to gravity.
• To derive a gravity force law that includes non-Newtonian terms through a modified Einstein-Hilbert action.

Proposed method

• Modify the Einstein-Hilbert action by introducing anisotropic terms to account for directional dependence in spacetime geometry.
• Derive generalized geodesic equations from the modified action principle, leading to non-standard curvature effects.
• Reformulate the effective gravity force law to include additional terms beyond Newtonian gravity, arising from the anisotropic metric structure.
• Ensure the modified theory remains consistent with the equivalence principle by preserving local Lorentz invariance in the weak-field limit.
• Apply the resulting equations to model galactic rotation curves and gravitational potential wells in 3D space.
• Compare predictions with observational data, including the Tully-Fisher relation, to validate the model.

Experimental results

Research questions

• RQ1 Can anisotropic corrections to the Einstein-Hilbert action explain flat rotation curves in spiral galaxies without invoking dark matter?
• RQ2 How does the modified gravity force law derived from the generalized action differ from Newtonian gravity in spherically symmetric and rotating systems?
• RQ3 To what extent does the modified geometrodynamics reproduce the Tully-Fisher relation for spiral galaxies?
• RQ4 Does the inclusion of anisotropy preserve the equivalence principle in the weak-field limit?
• RQ5 Can the 3D gravitational well of spiral galaxies be consistently explained using this modified geometric framework?

Key findings

• The modified action leads to generalized geodesic equations that naturally incorporate non-Newtonian contributions to gravity.
• The resulting gravity force law includes additional terms that reproduce flat rotation curves without requiring dark matter.
• The model successfully accounts for the 3D gravitational well depth observed in spiral galaxies.
• The Tully-Fisher relation emerges as a direct consequence of the modified dynamics, consistent with observations.
• The theory maintains compatibility with the equivalence principle, ensuring consistency with foundational principles of relativity.
• All standard results of classical geometrodynamics are preserved in the weak-field and isotropic limits.