On Topological Modifications of Newton's Law

TL;DR

This paper explores how a global R^2 x S^1 topology of space modifies Newton's law, leading to anisotropic corrections at small scales and a logarithmic potential at large scales, with implications for cosmology and galactic dynamics.

Contribution

It introduces a simple topological model that alters Newton's law, deriving explicit corrections and analyzing their effects on cosmic acceleration and galactic rotation.

Findings

At small distances, Newton's potential gains multipole anisotropic corrections.

At large distances, the potential becomes logarithmic with small exponential corrections.

The topology change implies a weakening of cosmic deceleration and measurable effects on galactic orbits.

Abstract

Recent cosmological data for very large distances challenge the validity of the standard cosmological model. Motivated by the observed spatial flatness the accelerating expansion and the various anisotropies with preferred axes in the universe we examine the consequences of the simple hypothesis that the three-dimensional space has a global R^2 X S^1 topology. We take the radius of the compactification to be the observed cosmological scale beyond which the accelerated expansion starts. We derive the induced corrections to the Newton's gravitational potential and we find that for distances smaller than the S^1-radius the leading 1/r-term is corrected by convergent power series of multipole form in the polar angle making explicit the induced anisotropy by the compactified third dimension. On the other hand, for distances larger than the compactification scale the asymptotic behavior of the potential exhibits a logarithmic dependence with exponentially small corrections. The change of Newton's force from 1/r^2 to 1/r behavior implies a weakening of the deceleration for the expanding universe. Such topologies can also be created locally by standard Newtonian axially symmetric mass distributions with periodicity along the symmetry axis. In such cases we can use our results to obtain measurable modifications of Newtonian orbits for small distances and flat rotation spectra, for large distances at the galactic level.