Gravitational Interaction in the Chimney Lattice Universe

TL;DR

This paper explores how a specific chimney topology of the universe affects gravitational potential and force calculations, presenting three methods with practical advantages for computational efficiency.

Contribution

It introduces three distinct solutions for gravitational potential in a chimney universe topology, highlighting the most computationally efficient method.

Findings

Yukawa potential summation is most efficient for current universe calculations.

Three different mathematical approaches to solve the gravitational potential.

Plain Yukawa summation requires fewer terms for accurate results.

Abstract

We investigate the influence of the chimney topology $T\times T\times R$ of the Universe on the gravitational potential and force that are generated by point-like massive bodies. We obtain three distinct expressions for the solutions. One follows from Fourier expansion of delta functions into series using periodicity in two toroidal dimensions. The second one is the summation of solutions of the Helmholtz equation, for a source mass and its infinitely many images, which are in the form of Yukawa potentials. The third alternative solution for the potential is formulated via the Ewald sums method applied to Yukawa-type potentials. We show that, for the present Universe, the formulas involving plain summation of Yukawa potentials are preferable for computational purposes, as they require a smaller number of terms in the series to reach adequate precision.